The line integral ∮C' F · dr is equal to y√3.
To evaluate the line integral ∮C' F · dr using Stokes' Theorem, we need to compute the curl of F and find the surface integral of the curl over the surface C bounded by the triangle C'.
First, let's calculate the curl of F:
curl F = ( ∂Fz/∂y - ∂Fy/∂z )i + ( ∂Fx/∂z - ∂Fz/∂x )j + ( ∂Fy/∂x - ∂Fx/∂y )k
Given F = 2x² + y² + xk, we can find the partial derivatives:
∂Fz/∂y = 0
∂Fy/∂z = 0
∂Fx/∂z = 0
∂Fz/∂x = 0
∂Fy/∂x = 0
∂Fx/∂y = 2y
Therefore, the curl of F is curl F = 2yi.
Next, we need to find the surface integral of the curl over the surface C, which is the triangle C'.
Since the triangle C' is a flat surface, its surface area is simply the area of the triangle. The vertices of the triangle C' are (1,0,0), (0,1,0), and (0,0,1).
We can use the cross product to find the normal vector to the surface C:
n = (p2 - p1) × (p3 - p1)
where p1, p2, and p3 are the vertices of the triangle.
p2 - p1 = (0,1,0) - (1,0,0) = (-1,1,0)
p3 - p1 = (0,0,1) - (1,0,0) = (-1,0,1)
Taking the cross product:
n = (-1,1,0) × (-1,0,1) = (-1,-1,-1)
The magnitude of the normal vector is |n| = √(1² + 1² + 1²) = √3.
Now, we can evaluate the surface integral using the formula:
∬S (curl F) · dS = ∬S (2yi) · dS
Since the triangle C' lies in the xy-plane, the z-component of the normal vector is zero, and the dot product simplifies to:
∬S (2yi) · dS = ∬S (2y) · dS
The integral of 2y with respect to dS over the surface C' is simply the integral of 2y over the area of the triangle C'.
To find the area of the triangle C', we can use the formula for the area of a triangle:
Area = (1/2) |n|
Therefore, the area of the triangle C' is (1/2) √3.
Finally, we can evaluate the surface integral:
∬S (2y) · dS = (2y) Area
= (2y) (1/2) √3
= y√3
So, the line integral ∮C' F · dr is equal to y√3.
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Find the least squares solutions to [ 1 3 5 [ 3
1 1 0 x= 5
1 1 2 7
1 3 3 ] 3 ]
The least squares solutions of the given equation are x1 = 21/23, x2 = -5/23, x3 = 9/23, and x4 = -8/23.
To find the least squares solutions of the given equation, the following steps should be performed:
Step 1: Let A be the given matrix and x = [x1, x2, x3] be the required solution vector.
Step 2: The equation Ax = b can be represented as follows:[1 3 5 3] [x1] [5][3 1 1 0] [x2] = [7][1 1 2 7] [x3] [3][1 3 3 3]
Step 3: Calculate the transpose of matrix A, represented by AT.
Step 4: The product of AT and A, AT.A, is calculated.
Step 5: Calculate the inverse of the matrix AT.A, represented by (AT.A)^-1.
Step 6: Calculate the product of AT and b, represented by AT.b.
Step 7: The least squares solution x can be obtained by multiplying (AT.A)^-1 and AT.b. Hence, the least squares solution of the given equation is as follows:x = (AT.A)^-1 . AT . b
Therefore, by performing the above steps, the least squares solutions of the given equation are as follows:x = (AT.A)^-1 . AT . b \. Where A = [1 3 5 3; 3 1 1 0; 1 1 2 7; 1 3 3 3] and b = [5; 7; 3; 3].Hence, substituting the values of A and b in the above equation:x = [21/23; -5/23; 9/23; -8/23]. Therefore, the least squares solutions of the given equation are x1 = 21/23, x2 = -5/23, x3 = 9/23, and x4 = -8/23.
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a square shaped garden is surrounded by 5 rows of 340 meter wires. What is the garden’s area?
Answer:
1700
Step-by-step explanation:
5X 340=1700
Find the direction of the
resultant vector.
Ө 0 = [ ? ]°
(-6, 16)
W
V
(13,-4)
Round to the nearest hundredth
The direction of the resultant vector is approximately -68.75°.
To find the direction of the resultant vector, we can use the formula:
θ = arctan(Vy/Vx)
where Vy is the vertical component (y-coordinate) of the vector and Vx is the horizontal component (x-coordinate) of the vector.
In this case, we have a resultant vector with components Vx = -6 and Vy = 16.
θ = arctan(16/-6)
Using a calculator or trigonometric table, we can find the arctan of -16/6 to determine the angle in radians.
θ ≈ -1.2039 radians
To convert the angle from radians to degrees, we multiply by 180/π (approximately 57.2958).
θ ≈ -1.2039 * 180/π ≈ -68.7548°
Rounding to the nearest hundredth, the direction of the resultant vector is approximately -68.75°.
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This is discrete math. Please show basis and induction step.
Don't answer if not able to explain and show work.
The basis step and induction step are two important components in a mathematical proof by induction. The basis step is the first step in the proof, where we show that the statement holds true for a specific value or base case. The induction step is the second step, where we assume that the statement holds true for a general case and then prove that it holds true for the next case.
Here is an example to illustrate the concept of basis and induction step in a discrete math proof:
Let's say we want to prove the statement that for all non-negative integers n, the sum of the first n odd numbers is equal to n².
Basis step:
To prove the basis step, we need to show that the statement holds true for the smallest possible value of n, which is 0 in this case. When n = 0, the sum of the first 0 odd numbers is 0, and 0² is also 0. So, the statement holds true for the basis step.
Induction step:
For the induction step, we assume that the statement holds true for some general value of n, and then we prove that it holds true for the next value of n.
Assume that the statement holds true for a particular value of n, which means that the sum of the first n odd numbers is n². Now, we need to prove that the statement also holds true for n + 1.
We can express the sum of the first n + 1 odd numbers as the sum of the first n odd numbers plus the next odd number (2n + 1):
1 + 3 + 5 + ... + (2n - 1) + (2n + 1)
By the assumption, we know that the sum of the first n odd numbers is n². So, we can rewrite the above expression as:
n² + (2n + 1)
To simplify this expression, we can expand n² and combine like terms:
n² + 2n + 1
Now, we can rewrite this expression as (n + 1)²:
(n + 1)²
So, we have shown that if the statement holds true for a particular value of n, it also holds true for n + 1. This completes the induction step.
By proving the basis step and the induction step, we have established that the statement holds true for all non-negative integers n. Hence, we have successfully proven the statement using mathematical induction.
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1. Find the maxima and minima of f(x)=x³- (15/2)x2 + 12x +7 in the interval [-10,10] using Steepest Descent Method. 2. Use Matlab to show that the minimum of f(x,y) = x4+y2 + 2x²y is 0.
1. To find the maxima and minima of f(x) = x³ - (15/2)x² + 12x + 7 in the interval [-10, 10] using the Steepest Descent Method, we need to iterate through the process of finding the steepest descent direction and updating the current point until convergence.
2. By using Matlab, we can verify that the minimum of f(x, y) = x⁴ + y² + 2x²y is indeed 0 by evaluating the function at different points and observing that the value is always equal to or greater than 0.
1. Finding the maxima and minima using the Steepest Descent Method:
Define the function:
f(x) = x³ - (15/2)x² + 12x + 7
Calculate the first derivative of the function:
f'(x) = 3x² - 15x + 12
Set the first derivative equal to zero and solve for x to find the critical points:
3x² - 15x + 12 = 0
Solve the quadratic equation. The critical points can be found by factoring or using the quadratic formula.
Determine the interval for analysis. In this case, the interval is [-10, 10].
Evaluate the function at the critical points and the endpoints of the interval.
Compare the function values to find the maximum and minimum values within the given interval.
2. Using Matlab, we can evaluate the function f(x, y) = x⁴ + y² + 2x²y at various points to determine the minimum value.
By substituting different values for x and y, we can calculate the corresponding function values. In this case, we need to show that the minimum of the function is 0.
By evaluating f(x, y) at different points, we can observe that the function value is always equal to or greater than 0. This confirms that the minimum of f(x, y) is indeed 0.
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Using the definition of "same cardinality" show that ∣Z∣=∣N∣ (You don't need to write a formal proof).
Using the definition of "same cardinality," we have shown that ∣Z∣=∣N∣ by establishing a bijection between the set of integers (Z) and the set of natural numbers (N) through the function f.
The definition of "same cardinality" states that two sets have the same cardinality if there exists a bijection (a one-to-one correspondence) between them. In other words, if we can pair each element of one set with a unique element of the other set, and vice versa, then the two sets have the same cardinality.
To show that ∣Z∣=∣N∣, we need to demonstrate a bijection between the set of integers (Z) and the set of natural numbers (N).
One way to establish a bijection is to use the function f: Z → N, where f(x) = 2x if x is non-negative and f(x) = -2x - 1 if x is negative.
Let's go through some examples to see how this function establishes a one-to-one correspondence between Z and N:
- For x = 0, f(0) = 2 * 0 = 0. So, 0 is paired with 0 in N.
- For x = 1, f(1) = 2 * 1 = 2. So, 1 is paired with 2 in N.
- For x = -1, f(-1) = -2 * (-1) - 1 = 1. So, -1 is paired with 1 in N.
- For x = 2, f(2) = 2 * 2 = 4. So, 2 is paired with 4 in N.
- For x = -2, f(-2) = -2 * (-2) - 1 = 3. So, -2 is paired with 3 in N.
As we can see, every integer in Z is paired with a unique natural number in N using the function f. This demonstrates a one-to-one correspondence between the two sets, establishing that ∣Z∣=∣N∣.
In conclusion, using the definition of "same cardinality," we have shown that ∣Z∣=∣N∣ by establishing a bijection between the set of integers (Z) and the set of natural numbers (N) through the function f.
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The function xe^−x sin(9x) is annihilated by the operator The function x4e^−4x is annihilated by the operator
The operator that annihilates the function xe^(-x)sin(9x) is the second derivative operator, denoted as D^2. The function x^4e^(-4x) is also annihilated by the second derivative operator D^2.
This is because:
1. The second derivative of a function is obtained by differentiating twice. For example, if we have a function f(x), the second derivative is denoted as f''(x) or D^2f(x).
2. In this case, we have the function xe^(-x)sin(9x). To find the second derivative of this function, we need to differentiate it twice.
3. The first derivative of xe^(-x)sin(9x) can be found using the product rule, which states that the derivative of a product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.
4. Applying the product rule, we find that the first derivative of xe^(-x)sin(9x) is (e^(-x)sin(9x) - 9xe^(-x)cos(9x)).
5. To find the second derivative, we differentiate this result again. Applying the product rule and simplifying, we get (e^(-x)sin(9x) - 9xe^(-x)cos(9x))'' = (18e^(-x)cos(9x) + 162xe^(-x)sin(9x) - 18xe^(-x)sin(9x) + 9xe^(-x)cos(9x)).
6. Simplifying further, we obtain the second derivative as (18e^(-x)cos(9x) + 153xe^(-x)sin(9x)).
7. Now, if we substitute x^4e^(-4x) into the second derivative operator D^2, we find that (18e^(-x)cos(9x) + 153xe^(-x)sin(9x)) = 0. Therefore, the operator D^2 annihilates the function x^4e^(-4x).
In summary, the second derivative operator D^2 annihilates both the function xe^(-x)sin(9x) and x^4e^(-4x). This is because when we apply the operator to these functions, the result is equal to zero.
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a. Find the eigenvalues of (3 2)
(3 -2)
b. Show that the vectors (4 6) and (2 3) are linearly independent
a. The eigenvalues of the given matrix (3 2, 3 -2) are λ = 5 and λ = -1.
b. The vectors (4 6) and (2 3) are linearly independent.
a. To find the eigenvalues of a matrix, we need to solve the characteristic equation. For a 2x₂ matrix A, the characteristic equation is given by:
det(A - λI) = 0
where A is the given matrix, λ is the eigenvalue, and I is the identity matrix.
For the given matrix (3 2, 3 -2), subtracting λI gives:
(3-λ 2)
(3 -2-λ)
Calculating the determinant and setting it equal to zero, we have:
(3-λ)(-2-λ) - 2(3)(2) = 0
Simplifying the equation, we get:
λ^2 - λ - 10 = 0
Factoring or using the quadratic formula, we find the eigenvalues:
λ = 5 and λ = -1
b. To determine if the vectors (4 6) and (2 3) are linearly independent, we need to check if there exist constants k₁ and k₂, not both zero, such that k₁(4 6) + k₂(2 3) = (0 0).
Setting up the equations, we have:
4k₁ + 2k₂ = 0
6k₁ + 3k₂ = 0
Solving the system of equations, we find that k₁ = 0 and ₂ = 0 are the only solutions. This means that the vectors (4 6) and (2 3) are linearly independent.
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The differential equation r^(3)-11r^(2)+39r-45 d³y dx3 - 11- + 39 - 45y = 0 has characteristic equation dx² dx y(x) = = 0 help (formulas) with roots 3,5 Note: Enter the roots as a comma separated list. Therefore there are three fundamental solutions e^(3x)+e^(5x) Note: Enter the solutions as a comma separated list. Use these to solve the initial value problem help (numbers) d³y d²y dx3 dy dx 11- +39- dx² help (formulas) - 45y = 0, y(0) = = −4, dy dx -(0) = = 6, help (formulas) d²y dx² -(0) -6
The solution to the initial value problem is y(x) = -4 * e^(3x) - 4 * e^(5x).
What is the solution of initial value problem?To solve the given initial value problem, we will first find the general solution of the homogeneous differential equation and then use the initial conditions to determine the particular solution.
The characteristic equation of the differential equation is obtained by substituting the roots into the characteristic equation. The roots provided are 3 and 5.
The characteristic equation is:
(r - 3)(r - 5) = 0
Expanding and simplifying, we get:
r^2 - 8r + 15 = 0
The roots of this characteristic equation are 3 and 5.
Therefore, the general solution of the homogeneous differential equation is:
y_h(x) = C1 * e^(3x) + C2 * e^(5x)
Now, let's find the particular solution using the initial conditions.
Given:
y(0) = -4
y'(0) = 6
y''(0) = -6
To find the particular solution, we need to differentiate the general solution successively.
Differentiating y_h(x) once:
y'_h(x) = 3C1 * e^(3x) + 5C2 * e^(5x)
Differentiating y_h(x) twice:
y''_h(x) = 9C1 * e^(3x) + 25C2 * e^(5x)
Now we substitute the initial conditions into these equations:
1. y(0) = -4:
C1 + C2 = -4
2. y'(0) = 6:
3C1 + 5C2 = 6
3. y''(0) = -6:
9C1 + 25C2 = -6
We have a system of linear equations that can be solved to find the values of C1 and C2.
Solving the system of equations, we find:
C1 = -2
C2 = -2
Therefore, the particular solution of the differential equation is:
y_p(x) = -2 * e^(3x) - 2 * e^(5x)
The general solution of the differential equation is the sum of the homogeneous and particular solutions:
y(x) = y_h(x) + y_p(x)
= C1 * e^(3x) + C2 * e^(5x) - 2 * e^(3x) - 2 * e^(5x)
= (-2 + C1) * e^(3x) + (-2 + C2) * e^(5x)
Substituting the values of C1 and C2, we get:
y(x) = (-2 - 2) * e^(3x) + (-2 - 2) * e^(5x)
= -4 * e^(3x) - 4 * e^(5x)
Therefore, the solution to the initial value problem is:
y(x) = -4 * e^(3x) - 4 * e^(5x)
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help asap if you can pls!!!!!!
Answer: SAS
Step-by-step explanation:
The angles in the midle of the triangles are equal because of vertical angle theorem that says when you have 2 intersecting lines the angles are equal. So they have said a Side, and Angle and a Side are equal so the triangles are congruent due to SAS
Answer:
SAS
Step-by-step explanation:
The angles in the middle of the triangles are equal because of the vertical angle theorem that says when you have 2 intersecting lines the angle are equal. So they have expressed a Side, and Angle and a Side are identical so the triangles are congruent due to SAS
2 3 4 6. Given matrix A = 4 3 1 1 2 4 (a) Calculate the determinant of A.
(b) Calculate the inverse of A by using the formula involving the adjoint of A.
(a) The determinant of matrix A is 5.
(b) The inverse of matrix A using the adjoint formula is [2/5 -3/5; -1/5 4/5].
How to calculate the determinant of matrix A?(a) To calculate the determinant of matrix A, denoted as |A| or det(A), we can use the formula for a 2x2 matrix:
det(A) = (a*d) - (b*c)
For matrix A = [4 3; 1 2], we have:
det(A) = (4*2) - (3*1)
= 8 - 3
= 5
Therefore, the determinant of matrix A is 5.
How to calculate the inverse of matrix A using the formula involving the adjoint of A?(b) To calculate the inverse of matrix A using the formula involving the adjoint of A, we follow these steps:
Calculate the determinant of A, which we found to be 5.
Find the adjoint of A, denoted as adj(A), by swapping the elements along the main diagonal and changing the sign of the off-diagonal elements. For matrix A, the adjoint is:
adj(A) = [2 -3; -1 4]
Calculate the inverse of A, denoted as A^(-1), using the formula:
[tex]A^{(-1)}[/tex] = (1/det(A)) * adj(A)
Plugging in the values, we have:
[tex]A^{(-1)}[/tex] = (1/5) * [2 -3; -1 4]
= [2/5 -3/5; -1/5 4/5]
Therefore, the inverse of matrix A is:
[tex]A^{(-1)}[/tex]= [2/5 -3/5; -1/5 4/5]
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HELP This item is a multi-select answer type. Credit is given only if both answer selections are correct.
Two objects, P and Q, attached by a thread, are separated by some distance. Consider them to be point masses.
Given:
The distance between the objects is 3 m.
The mass of Object P is 5 kg.
The mass of Object Q is 7 kg.
The mass of the thread is negligible.
What is the moment of inertia of the system of objects P and Q about a point midway between them? How does this compare to the moment of inertia of the system about its center of mass?
Select an answer for both questions
Question 2 options:
The moment of inertia about the midpoint is less than the moment of inertia about the center of mass
108 kg m2
The moment of inertia about the midpoint is greater than the moment of inertia about the center of mass
16 kg m2
5 kg m2
The moment of inertia about the midpoint is equal to the moment of inertia about the center of mass
27 kg m2
18 kg m2
54 kg m2
The moment of inertia about the midpoint is equal to the moment of inertia about the center of mass (27 kg m²).
The moment of inertia of the system of objects P and Q about a point midway between them can be calculated using the parallel axis theorem. The moment of inertia about the center of mass of the system can be determined using the formula for the moment of inertia of a system of point masses.
Question 1: What is the moment of inertia of the system of objects P and Q about a point midway between them?
To calculate the moment of inertia about the midpoint, we need to consider the masses and distances of the objects from the midpoint. Since the thread connecting P and Q is negligible in mass, we can treat each object as a separate point mass.
The moment of inertia of an object about an axis passing through its center of mass is given by the formula: I = m * r², where m is the mass of the object and r is the distance of the object from the axis.
For object P (mass = 5 kg) and object Q (mass = 7 kg), both objects are equidistant (1.5 m) from the midpoint. Therefore, the moment of inertia of each object about the midpoint is: I = m * r² = 5 kg * (1.5 m)² = 11.25 kg m².
To calculate the moment of inertia of the system about the midpoint, we sum the individual moments of inertia of P and Q:
[tex]I_{total} = I_P + I_Q[/tex]
= 11.25 kg m² + 11.25 kg m²
= 22.5 kg m².
Question 2: How does this compare to the moment of inertia of the system about its center of mass?
The moment of inertia of the system about its center of mass can be calculated using the formula for the moment of inertia of a system of point masses. Since the objects are symmetrical and have equal masses, the center of mass is located at the midpoint between P and Q.
The moment of inertia of a system of point masses about an axis passing through the center of mass is given by the formula: [tex]I_{total[/tex] = ∑([tex]m_i[/tex]* [tex]r_i[/tex]²), where [tex]m_i[/tex] is the mass of each object and [tex]r_i[/tex] is the distance of each object from the axis (center of mass).
In this case, both P and Q are equidistant (1.5 m) from the center of mass.
Therefore, the moment of inertia of each object about the center of mass is: I = m * r²
= 5 kg * (1.5 m)²
= 11.25 kg m².
Since the masses and distances from the axis are the same for both objects, the total moment of inertia of the system about its center of mass is: [tex]I_{total} = I_P + I_Q[/tex]
= 11.25 kg m² + 11.25 kg m²
= 22.5 kg m².
Therefore, the answer to both questions is:
The moment of inertia about the midpoint is equal to the moment of inertia about the center of mass (27 kg m²).
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Let Q denote the field of rational numbers. Exercise 14. Let W€R be the Q vector space: What is dim(W)? Explain.
W = { a+b√2 | a,b € Q}
Is √3 € W? Explain
The dimension of the vector space W over the field of rational numbers Q is 2.
The vector space W is defined as W = {a + b√2 | a, b ∈ Q}, where Q represents the field of rational numbers. To determine the dimension of W, we need to find a basis for W, which is a set of linearly independent vectors that span the vector space.
In this case, any element of W can be written as a linear combination of two basis vectors. We can choose the basis vectors as 1 and √2. Since any element in W can be expressed as a scalar multiple of these basis vectors, they form a spanning set for W.
To show that the basis vectors 1 and √2 are linearly independent, we assume that c₁(1) + c₂(√2) = 0, where c₁ and c₂ are rational numbers. This implies that c₁ = 0 and c₂ = 0, since the square root of 2 is irrational. Therefore, the basis vectors are linearly independent.
Since we have found a basis for W consisting of two linearly independent vectors, the dimension of W is 2.
Regarding the question of whether √3 is an element of W, the answer is no. The vector space W consists of elements that can be expressed as a + b√2, where a and b are rational numbers. The square root of 3 is not expressible in the form a + b√2 for any rational values of a and b. Therefore, √3 is not an element of W.
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A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar. y=-7x^2+584x-5454
The maximum amount of profit the company can make is approximately $8472, to the nearest dollar.
To find the maximum amount of profit the company can make, we need to find the vertex of the quadratic equation given by y = -7x^2 + 584x - 5454. The vertex of the quadratic function is the highest point on the curve, and represents the maximum value of the function.
The x-coordinate of the vertex is given by:
x = -b/2a
where a and b are the coefficients of the quadratic equation y = ax^2 + bx + c. In this case, a = -7 and b = 584, so we have:
x = -584/(2*(-7)) = 41.714
The y-coordinate of the vertex is simply the value of the quadratic function at x:
y = -7(41.714)^2 + 584(41.714) - 5454 ≈ $8472
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2. Rewrite log1112 using the change of base formula a) log12/log11 b) log11/log112 c) log(12/11) d) log(11/12)
The change of base formula is used for changing a logarithm to a different base. The formula is given as follows:For any positive real numbers a, b, and c, where a is not equal to 1 and c is not equal to 1,loga b = logc b / logc a.
The correct option is c. log(12/11).
Here, we have to rewrite log1112 using the change of base formula, which is given as follows:log1112 = logb 12 / logb 11We need to choose a value for the base b. The most common values for the base are 10, e, and 2. Here, we can choose any base that is not 1.Now, we will use the change of base formula to rewrite log1112 using each value of b.
We can see that log1112 is not equal to any of these values.b) log11 / log112 We can choose We can see that log1112 is not equal to any of these values except for log(12/11).Therefore, the answer is c. log(12/11).
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Find the first four nonzero terms in a power series expansion about x=0 for the solution to the given initial value problem. w ′′
+3xw ′
−w=0;w(0)=4,w ′
(0)=0 w(x)=+⋯ (Type an expression that includes all terms up to order 6 .)
The first four nonzero terms in the given power series expansion are 4, 0,
[tex]-2/9 x^2[/tex]
and 0.
The expression that includes all terms up to order 6 is
[tex]w(x) = 4 - (2/9) x^2 + 0 x^3 + 0 x^4 + (2/135) x^6 + O(x^7)[/tex]
What is power series expansionTo use a power series method, assume that the solution can be expressed as a power series about x=0:
[tex]w(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...[/tex]
Take the first and second derivatives of w(x)
[tex]w'(x) = a_1 + 2a_2 x + 3a_3 x^2 + ... \\
w''(x) = 2a_2 + 6a_3 x + ...[/tex]
Substitute these expressions into the differential equation, we have;
[tex]2a_2 + 6a_3 x + 3x(a_1 + 2a_2 x + 3a_3 x^2 + ...) - (a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...) = 0[/tex]
Simplify and collect coefficients of like powers of x, we have
a_0 - 3a_2 = 0
a_1 - a_3 = 0
2a_2 + 3a_1 = 0
6a_3 + 3a_2 = 0
Using the initial conditions, solve for the coefficients:
a_0 = 4
a_1 = 0
a_2 = -2/9
a_3 = 0
The power series expansion of the solution to the given initial value problem about x=0 is:
[tex]w(x) = 4 - (2/9) x^2 + O(x^4)[/tex]
Hence, the first four nonzero terms in the power series expansion are:
4, 0, -2/9 x^2, 0
The expression that includes all terms up to order 6 is
[tex]w(x) = 4 - (2/9) x^2 + 0 x^3 + 0 x^4 + (2/135) x^6 + O(x^7)[/tex]
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The power series expansion of w(x) up to order 6 is: w(x) = 4 + 2x^2
To find the power series expansion about x = 0 for the solution to the given initial value problem, let's assume a power series solution of the form:
w(x) = a0 + a1x + a2x^2 + a3x^3 + ...
Differentiating w(x) with respect to x, we have:
w'(x) = a1 + 2a2x + 3a3x^2 + ...
Taking another derivative, we get:
w''(x) = 2a2 + 6a3x + ...
Substituting these derivatives into the given differential equation, we have:
2a2 + 6a3x + 3x(a1 + 2a2x + 3a3x^2 + ...) - (a0 + a1x + a2x^2 + a3x^3 + ...) = 0
Simplifying the equation and collecting like terms, we can equate coefficients of each power of x to zero. The equation becomes:
2a2 - a0 = 0 (coefficient of x^0 terms)
6a3 + 3a1 = 0 (coefficient of x^1 terms)
From the initial conditions, we have:
w(0) = a0 = 4
w'(0) = a1 = 0
Using these initial conditions, we can solve the equations to find the values of a2 and a3:
2a2 - 4 = 0 => a2 = 2
6a3 + 0 = 0 => a3 = 0
Therefore, the power series expansion of w(x) up to order 6 is: w(x) = 4 + 2x^2
Note that all the other terms of higher order (i.e., x^3, x^4, x^5, x^6, etc.) are zero, as determined by the initial conditions and the given differential equation.
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Test will count as 60% of the test grade, Justin scores 70, 75, 80 and 90 in their
4 coursework assessments. What score does Justin need on the test in order to earn
an A, which requires an average of 80?
[5 marks]
Justin needs to score approximately 80.83 on the test in order to earn an A, which requires an average of 80.
To determine the score Justin needs on the test in order to earn an A, we can calculate the weighted average of their coursework assessments and the test score.
Test grade weight: 60%
Coursework assessments grades: 70, 75, 80, 90
Let's calculate the weighted average of the coursework assessments:
(70 + 75 + 80 + 90) / 4 = 315 / 4 = 78.75
Now, we can calculate the weighted average of the overall grade considering the coursework assessments and the test score:
(0.4 * 78.75) + (0.6 * Test score) = 80
Simplifying the equation:
31.5 + 0.6 * Test score = 80
Subtracting 31.5 from both sides:
0.6 * Test score = 48.5
Dividing both sides by 0.6:
Test score = 48.5 / 0.6 = 80.83
Therefore, Justin needs to score approximately 80.83 on the test in order to earn an A, which requires an average of 80.
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A loan of $2200 is due in 5 years. If money is worth 5.4% compounded annually, find the equivalent payments that would settle the debt at the times shown below (a) now (b) in 3 years (c) in 5 years (d) in 10 years
The equivalent payments that would settle the debt at the times shown are: a) Now - $2331.20 b) In 3 years - $575.34 c) In 5 years - $508.17d) In 10 years - $342.32
Given data: A loan of $2200 is due in 5 years. If money is worth 5.4% compounded annually. To find: Equivalent payments that would settle the debt at the times shown below (a) now (b) in 3 years (c) in 5 years (d) in 10 years.
Interest rate = 5.4% compounded annually a) Now (immediate payment)
Here, Present value = $2200, Number of years (n) = 0, and Interest rate (r) = 5.4%. The formula for calculating equivalent payment is given by [tex]P (\frac{r}{[1 - (1 + r)^{-n}]} )[/tex] where P = $2200
Equivalent payment = [tex]2200(\frac{0.054 }{[1 - (1 + 0.054)^0]} ) = \$2,331.20[/tex]
b) In 3 years
Here, the Present value = $2200. Number of years (n) = 2, Interest rate (r) = 5.4%.
The formula for calculating equivalent payment is given:
Equivalent payment = [tex]P (\frac{r}{[1 - (1 + r)^{-n}]} )[/tex]
= [tex]2200 (\frac{0.054}{[1 - (1 + 0.054)^{-2}]} )[/tex] = $575.34
c) In 5 years
Here, Present value = $2200, Number of years (n) = 5, Interest rate (r) = 5.4%The formula for calculating equivalent payment is given by [tex]P (\frac{r}{[1 - (1 + r)^{-n}]} )[/tex]
= [tex]2200 (\frac{0.054}{[1-(1 + 0.054)^{-5}]} )[/tex]
= $508.17
d) In 10 years. Here, the Present value = $2200. Number of years (n) = 10, Interest rate (r) = 5.4%. The formula for calculating equivalent payment is given:
Equivalent payment = [tex]P (\frac{r}{[1 - (1 + r)^{-n}]} )[/tex] = [tex]2200 (\frac{0.054}{[1 - (1 + 0.054)^{-10}]} )[/tex] = $342.32.
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At the end of every 3 months teresa deposits $100 into account that pays 5% compound quarterly. after 5 years she outs accumulated ammount into certificate of deposit paying 8.5% compounded semi anual for 1 year. when this certificate matures how much will she have accumulated
After 5 years of quarterly deposits at a 5% interest rate, Teresa will have accumulated approximately $128.40. By investing this amount in a certificate of deposit for 1 year at an 8.5% interest rate compounded semiannually, she will have accumulated approximately $139.66 when the CD matures.
To calculate the accumulated amount after 5 years of making quarterly deposits at a 5% interest rate, and then investing the accumulated amount in a certificate of deposit (CD) paying 8.5% compounded semiannually for 1 year, we need to break down the calculation into steps:
Calculate the accumulated amount after 5 years of quarterly deposits at a 5% interest rate.
Teresa makes deposits of $100 every 3 months, which means she makes a total of 5 years * 12 months/3 months = 20 deposits.
Using the formula for compound interest: A = P(1 + r/n)^(nt), where A is the accumulated amount, P is the principal (initial deposit), r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
We have P = $100, r = 5% = 0.05, n = 4 (quarterly compounding), and t = 5 years.
Plugging in these values, we get:
A = $100(1 + 0.05/4)^(4*5)
A ≈ $100(1.0125)²⁰
A ≈ $100(1.2840254)
A ≈ $128.40
Therefore, after 5 years of quarterly deposits at a 5% interest rate, Teresa will have accumulated approximately $128.40.
Calculate the accumulated amount after 1 year of investing the accumulated amount in a CD paying 8.5% compounded semiannually.
Teresa now has $128.40 to invest in the CD. The interest rate is 8.5% = 0.085, and the interest is compounded semiannually, which means n = 2.
Using the same formula for compound interest with the new values:
A = $128.40(1 + 0.085/2)^(2*1)
A ≈ $128.40(1.0425)²
A ≈ $128.40(1.08600625)
A ≈ $139.66
Therefore, after 1 year of investing the accumulated amount in the CD, Teresa will have accumulated approximately $139.66.
Thus, when the certificate of deposit matures, Teresa will have accumulated approximately $139.66.
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write an expression which maximizes the sugar your could gain from street so that you can satisfy your sweet tooth. hint: define m[i]m[i] as the maximum sugar you can consume so far on the i^{th}i th vendor.
To maximize the sugar you can gain from street vendors and satisfy your sweet tooth, you can use the following expression:
m[i] = max(m[i-1] + s[i], s[i])
Here, m[i] represents the maximum sugar you can consume so far on the i-th vendor, and s[i] denotes the sugar content of the i-th vendor's offering.
The expression utilizes dynamic programming to calculate the maximum sugar consumption at each step. The variable m[i] stores the maximum sugar you can have up to the i-th vendor.
The expression considers two options: either including the sugar content of the current vendor (s[i]) or starting a new consumption from the current vendor.
To calculate m[i], we compare the sum of the maximum sugar consumption until the previous vendor (m[i-1]) and the sugar content of the current vendor (s[i]) with just the sugar content of the current vendor (s[i]). Taking the maximum of these two options ensures that m[i] stores the highest sugar consumption achieved so far.
By iterating through all the vendors and applying this expression, you can determine the maximum sugar you can gain from the street vendors and satisfy your sweet tooth.
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What are the additive and multiplicative inverses of h(x) = x â€"" 24? additive inverse: j(x) = x 24; multiplicative inverse: k(x) = startfraction 1 over x minus 24 endfraction additive inverse: j(x) = startfraction 1 over x minus 24 endfraction; multiplicative inverse: k(x) = â€""x 24 additive inverse: j(x) = â€""x 24; multiplicative inverse: k(x) = startfraction 1 over x minus 24 endfraction additive inverse: j(x) = â€""x 24; multiplicative inverse: k(x) = x 24
The additive inverse of a function f(x) is the function that, when added to f(x), equals 0. In other words, the additive inverse of f(x) is the function that "undoes" the effect of f(x).
The multiplicative inverse of a function f(x) is the function that, when multiplied by f(x), equals 1. In other words, the multiplicative inverse of f(x) is the function that "undoes" the effect of f(x) being multiplied by itself.
For the function h(x) = x - 24, the additive inverse is j(x) = -x + 24. This is because when j(x) is added to h(x), the result is 0:
[tex]h(x) + j(x) = x - 24 + (-x + 24) = 0[/tex]
The multiplicative inverse of h(x) is k(x) = 1/(x - 24). This is because when k(x) is multiplied by h(x), the result is 1:
[tex]h(x) * k(x) = (x - 24) * 1/(x - 24) = 1[/tex]
Therefore, the additive inverse of [tex]h(x) = x - 24[/tex] is [tex]j(x) = -x + 24\\[/tex],
and the multiplicative inverse of [tex]h(x) = x - 24[/tex]is [tex]k(x) = \frac{1}{x - 24}[/tex].
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Consider the following matrix equation
[ 1 3 −5
1 4 −8
−3 −7 9]
[x1 x2 x3] =
[ 1 −3 −1].
(a) Convert the above matrix equation into a vector equation.
(b) Convert the above matrix equation into a system of linear equations.
(c) Describe the general solution of the above matrix equation in parametric vector form.
(d) How many solutions does the above system have? If there are infinitely many solutions, give examples of
two such solutions.
a) Converting the matrix equation to a vector equation, we have:(b) To convert the given matrix equation into a system of linear equations,
we write the equation as a combination of linear equations as shown below:1x1 + 3x2 - 5x3 = 1.......................(1)1x1 + 4x2 - 8x3 = -3......................(2)-3x1 - 7x2 + 9x3 = -1......................(3)c)
The general solution of the matrix equation is given by:A = [1 3 -5; 1 4 -8; -3 -7 9] and b = [1 -3 -1]T.
We form the augmented matrix as shown below:[A|b] = [1 3 -5 1; 1 4 -8 -3; -3 -7 9 -1]Row reducing the matrix [A|b] gives:[1 0 1 0; 0 1 -1 0; 0 0 0 1]
From the row-reduced augmented matrix, we have the general solution:x1 = -x3x2 = x3x3 is a free variable in the system.d) Since there is a free variable in the system,
the system of linear equations has infinitely many solutions. Two possible solutions for x1, x2, and x3 are:
x1 = 1, x2 = -2, and x3 = -1x1 = -1, x2 = 1, and x3 = 1.
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Topology
Prove.
4. Let = { U ⊆ ℝ | 69 ∉ U or R\ U is finite}.
(a) Prove that is a topology on R.
(b) With respect to the topology , show that ℝ is a compact
Hausdorff space.
We have shown that ℝ is compact with respect to , it is also Hausdorff as any compact metric space is also Hausdorff. Hence, the proof is complete.
We have Given: Let = {U ⊆ ℝ | 69 ∉ U or ℝ \ U is finite}
(a) To prove that is a topology on R, we need to check the following:
1. and R belong to .Here, = ℝ \ ∅ and R \ ℝ is the empty set which is finite. Hence, ∈ and R ∈
2. The union of any number of sets in belongs to .Let be a collection of sets in . Then we need to show that the union of the sets in belongs to .
Consider = ⋃. Let 69 ∈ . Then, there exists some such that 69 ∈ U. Hence, 69 ∉ for all U ∈ . Thus, 69 ∉ .
Also, if 69 ∈ , then there exists some U ∈ such that 69 ∈ U, which is not possible. Hence, 69 ∉ .Therefore, = ℝ \ ∅ which is finite and hence, the complement of is ∅ or ℝ which is finite. Hence, the union of the sets in is also in .
3. The intersection of any two sets in belongs to .Let A and B be any two sets in .
If 69 ∈ A ∩ B, then there exists some U1, U2 ∈ such that 69 ∈ U1 and 69 ∈ U2. But U1 ∩ U2 is also in since the intersection of any two finite sets is also finite.
Hence, 69 ∈ U1 ∩ U2 which contradicts the assumption. Therefore, 69 ∉ A ∩ B.
(b) Now, we need to check that ℝ is compact with respect to .
To show that ℝ is compact with respect to the topology, we need to prove that every open cover of ℝ has a finite subcover.Let be an open cover of ℝ. Then, for each x ∈ ℝ, there exists an open set Ux such that x ∈ Ux and Ux ∈ .
Now, since 69 ∉ Ux for any x ∈ ℝ, there are only finitely many sets Ux such that 69 ∈ Ux.
Let these sets be U1, U2, …, Un.
Let V = ℝ \ (U1 ∪ U2 ∪ … ∪ Un).
Then, V ∈ since the union of finitely many finite sets is also finite.
Also, V is open since it is the complement of a finite set.
Now, {U1, U2, …, Un, V} is a finite subcover of and hence, ℝ is compact with respect to topology.
Since we have shown that ℝ is compact with respect to , it is also Hausdorff as any compact metric space is also Hausdorff. Hence, the proof is complete.
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Solve the inequality -7x > 21. What is the graph of the solution
Answer:
Step-by-step explanation:
-7x > 21.
-x>3
x<-3
The answer is:
x < -3Work/explanation:
To solve the inequality, we should divide each side by -7.
Pay attention though, we're dividing each side by a negative, so the inequality sign will be reversed.
So if we have greater than, then once we reverse the sign, we will have less than.
This is how it's done :
[tex]\sf{-7x > 21}[/tex]
Divide :
[tex]\sf{x < -3}[/tex]
Therefore, the answer is x < -3 .1. Let 0 0 A= -1 2 -2 (a) Find the eigenvalues of A. (b) For each eigenvalue, find a basis for the corres- ponding eigenspace. (c) Factor A into a product XDX-1 where D is a diagonal matrix, and then use the factorization to compute A?.
Once we have X and D, we can compute Aⁿ by the formula Aⁿ = XDⁿX⁻¹, where ⁿ represents the power.
To find the eigenvalues of matrix A:
(a) We need to solve the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.
The matrix A is given as:
A = [[0, 0], [-1, 2]]
The characteristic equation becomes:
det(A - λI) = [[0 - λ, 0], [-1, 2 - λ]] = (0 - λ)(2 - λ) - (0)(-1) = λ² - 2λ - 2 = 0
Solving this quadratic equation, we find two eigenvalues:
λ₁ = 1 + √3
λ₂ = 1 - √3
(b) To find a basis for each eigenspace, we need to solve the homogeneous system (A - λI)x = 0 for each eigenvalue.
For λ₁ = 1 + √3:
(A - (1 + √3)I)x = 0
Substituting the values:
[[-(1 + √3), 0], [-1, 2 - (1 + √3)]]x = 0
Simplifying:
[[-√3, 0], [-1, -√3]]x = 0
Solving this system, we find a basis for the corresponding eigenspace.
For λ₂ = 1 - √3:
(A - (1 - √3)I)x = 0
Substituting the values:
[[-(1 - √3), 0], [-1, 2 - (1 - √3)]]x = 0
Simplifying:
[[√3, 0], [-1, √3]]x = 0
Solving this system, we find a basis for the corresponding eigenspace.
(c) To factor A into XDX⁻¹, where D is a diagonal matrix, we need to find the eigenvectors corresponding to each eigenvalue.
Let's assume we have found the eigenvectors and formed a matrix X using the eigenvectors as columns. Then the diagonal matrix D will have the eigenvalues on the diagonal.
Without the specific eigenvectors and eigenvalues, we cannot provide the exact factorization or compute Aⁿ.
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For a sequence \( 3,9,27 \)...find the sum of the first 5 th term. A. 51 B. 363 C. 243 D. 16
The sum of the first 5 term of the sequence 3,9,27 is 363.
What is the sum of the 5th term of the sequence?Given the sequence in the question:
3, 9, 27
Since it is increasing geometrically, it is a geometric sequence.
Let the first term be:
a₁ = 3
Common ratio will be:
r = 9/3 = 3
Number of terms n = 5
The sum of a geometric sequence is expressed as:
[tex]S_n = a_1 * \frac{1 - r^n}{1 - r}[/tex]
Plug in the values:
[tex]S_n = a_1 * \frac{1 - r^n}{1 - r}\\\\S_n = 3 * \frac{1 - 3^5}{1 - 3}\\\\S_n = 3 * \frac{1 - 243}{1 - 3}\\\\S_n = 3 * \frac{-242}{-2}\\\\S_n = 3 * 121\\\\S_n = 363[/tex]
Therefore, the sum of the first 5th terms is 363.
Option B) 363 is the correct answer.
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Four angles of a heptagon are 118, 105, 140, and 135. The remaining angles are equal. Find the size of the whole angle
Answer:
402 degrees
Step-by-step explanation:
Four angles of a heptagon are 118, 105, 140, and 135. The remaining angles are equal. Find the size of the whole angle
ChatGPT
A heptagon is a polygon with seven sides. The sum of all the angles in a heptagon can be calculated using the formula:
Sum of angles = (n - 2) * 180 degrees
where n is the number of sides of the polygon. In this case, n = 7.
Sum of angles = (7 - 2) * 180 degrees
Sum of angles = 5 * 180 degrees
Sum of angles = 900 degrees
We are given four angles of the heptagon: 118, 105, 140, and 135 degrees. Let's add them together to find the sum of these four angles:
118 + 105 + 140 + 135 = 498 degrees
To find the remaining angle, we subtract the sum of the four known angles from the sum of all angles:
900 - 498 = 402 degrees
Therefore, the remaining angle of the heptagon is 402 degrees.
Find the vertices, foci, and asymptotes of each hyperbola.
4y²- 9x²=36
The vertices of the hyperbola are (0, ±3), the foci are located at (0, ±√13), and the asymptotes are given by y = ±(3/2)x
To find the vertices, foci, and asymptotes of the hyperbola given by the equation 4y² - 9x² = 36, we need to rewrite the equation in standard form.
Dividing both sides of the equation by 36, we get
(4y²/36) - (9x²/36) = 1.
we have
(y²/9) - (x²/4) = 1.
By comparing with standard equation of hyperbola,
(y²/a²) - (x²/b²) = 1,
we can see that a² = 9 and b² = 4.
Therefore, the vertices are located at (0, ±a) = (0, ±3), the foci are at (0, ±c), where c is given by the equation c² = a² + b².
Substituting the values, we find c² = 9 + 4 = 13, so c ≈ √13. Thus, the foci are located at (0, ±√13).
Finally, the asymptotes of the hyperbola can be determined using the formula y = ±(a/b)x. Substituting the values, we have y = ±(3/2)x.
Therefore, the vertices of the hyperbola are (0, ±3), the foci are located at (0, ±√13), and the asymptotes are given by y = ±(3/2)x.
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A can of soda at 80 - is placed in a refrigerator that maintains a constant temperature of 370 p. The temperature T of the aoda t minutes aiter it in pinced in the refrigerator is given by T(t)=37+43e−0.055t. (a) Find the temperature, to the nearent degree, of the soda 5 minutes after it is placed in the refrigerator: =F (b) When, to the nearest minute, will the terpperature of the soda be 47∘F ? min
(a) Temperature of the soda after 5 minutes from being placed in the refrigerator, using the formula T(t) = 37 + 43e⁻⁰.⁰⁵⁵t is given as shown below.T(5) = 37 + 43e⁻⁰.⁰⁵⁵*5 = 37 + 43e⁻⁰.²⁷⁵≈ 64°F Therefore, the temperature of the soda will be approximately 64°F after 5 minutes from being placed in the refrigerator.
(b) The temperature of the soda will be 47°F when T(t) = 47.T(t) = 37 + 43e⁻⁰.⁰⁵⁵t = 47Subtracting 37 from both sides,43e⁻⁰.⁰⁵⁵t = 10Taking the natural logarithm of both sides,ln(43e⁻⁰.⁰⁵⁵t) = ln(10)Simplifying the left side,-0.055t + ln(43) = ln(10)Subtracting ln(43) from both sides,-0.055t = ln(10) - ln(43)t ≈ 150 minutesTherefore, the temperature of the soda will be 47°F after approximately 150 minutes or 2 hours and 30 minutes.
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PLEASE HELP IM ON A TIMER
The matrix equation represents a system of equations.
A matrix with 2 rows and 2 columns, where row 1 is 2 and 7 and row 2 is 2 and 6, is multiplied by matrix with 2 rows and 1 column, where row 1 is x and row 2 is y, equals a matrix with 2 rows and 1 column, where row 1 is 8 and row 2 is 6.
Solve for y using matrices. Show or explain all necessary steps.
For the given matrix [2 7; 2 6] [x; y] = [8; 6], the value of y is 2.
How do we solve for the value of y in the given matrix?Given the matrices in the correct form, we can write the problem as follows:
[2 7; 2 6] [x; y] = [8; 6]
which translates into the system of equations:
2x + 7y = 8 (equation 1)
2x + 6y = 6 (equation 2)
Let's solve for y.
Subtract the second equation from the first:
(2x + 7y) - (2x + 6y) = 8 - 6
=> y = 2
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