Let's arrange the steps for solving the expression using the order of operations (also known as PEMDAS/BODMAS):
Subtract: (42-13) = 29
Multiply: 4 × 3 = 12
Exponentiation: 5² = 25
Multiply: 8 × 3 = 24
Divide: 24 ÷ 29 (result from step 1) = 0.8276 (rounded to 4 decimal places)
Add: 0.8276 (result from step 5) + 25 (result from step 3) = 25.8276 (rounded to 4 decimal places)
Add: 25.8276 (result from step 6) + 12 (result from step 2) = 37.8276 (rounded to 4 decimal places)
The steps should be performed in the following order:
Subtract -> Multiply -> Exponentiation -> Multiply -> Divide -> Add -> Add
So the steps should be performed in this order to evaluate the expression correctly.
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The graph of the function f(x) = 1/x+6 + 15 is a transformation of the graph of the function g(x) = 1/x by
The transformation of g(x) to f(x) is g(x) is shifted up 15 units and shifted left by 16 units
Describing the transformation of g(x) to f(x).From the question, we have the following parameters that can be used in our computation:
The functions f(x) and g(x)
Where, we have
f(x) = 1/(x + 16) + 15
g(x) = 1/x
From the above, we have
Horizontal difference = 16 - 0 = 16
Vertical difference = 15 - 0 = 15
This means that the transformation of g(x) to f(x) is g(x) is shifted up 15 units and shifted left by 16 units
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the 2nd term of an GP is 50more than the pth term and (n+1)term is 56 find the first term
Answer:
To determine the first term of the geometric progression, let's assume it is 'a'. From the given information, we know that:
Second Term = a + 50
(n + 1)th Term = a + 56
We also have the formula for the nth term of a geometric sequence:
nth term = a * r^(n-1)
where 'r' is the common ratio between consecutive terms. Since the difference between consecutive terms remains constant, i.e., 50 for the second term and 56 for the (n + 1)th term, we can set up the equation:
a + 50 = a * r^1
a + 56 = a * r^2
Solve the equations simultaneously:
a + 50 = a * r^1 => a - 50/r = 0
a + 56 = a * r^2 => a - 56/r = 0
Adding the two equations, we get:
2a - (50/r + 56/r) = 0
2a - (50 + 56)/r = 0
Simplifying the equation further, we obtain:
2a - (106/r) = 0
Now, solve for 'a':
2a - (106/r) = 0
2a = (106/r)
=> 2a = 106/r
The value of 'a' depends on the value of 'r', so we cannot determine its exact value without knowing 'r'. However, once we know 'r', we can calculate the value of 'a'.
logx+5 64 = 2
Please help!!
Solve and round to the nearest 100th
Explain how
WILL GIVE BRAINIEST FIRST ANSWER
The left-hand side of the equation represents the logarithm of 64 with base x, and it is not possible to find a real value of x that satisfies the equation.
solve the equation logₓ(64) + 5 = 2, we can follow these steps:
Step 1: Move the constant term to the other side of the equation:
logₓ(64) = 2 - 5
logₓ(64) = -3
Step 2: Rewrite the equation in exponential form:
x^(-3) = 64
Step 3: Rewrite 64 as a power of x:
x^(-3) = x^6
Step 4: Set the exponents equal to each other:
-3 = 6
The equation -3 = 6 is not true, which means there is no real solution to the given equation.
Therefore, the equation logₓ(64) + 5 = 2 has no solution in the real number system.
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A local dry-cleaning company bought new equipment and its estimated useful life is 4 years. Using the straight-line depreciation method, what is the rate of depreciation each year?
Answer:
$2,500 or 25%
Step-by-step explanation:
Let's use the same example:
Cost of Equipment = $10,000
Useful Life = 4 years
Depreciation Rate = (Annual Depreciation / Cost of Equipment) * 100
Annual Depreciation = Cost of Equipment / Useful Life
Annual Depreciation = $10,000 / 4 years
Annual Depreciation = $2,500
Depreciation Rate = ($2,500 / $10,000) * 100
Depreciation Rate = 0.25 * 100
Depreciation Rate = 25%
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Answer:
Lat A34,13,14) and E = [8] Insert or to make the statement trus
A \ mathbb E
Insert or to make the statement trun
A)-
(1)0$
A dairy needs 258 gallons of milk containing 7% butterfat how many gallons each of milk containing 8% butterfat and milk containing 2% butterfat must be used to obtain the desired 258 gallons
Let's assume x gallons of milk containing 8% butterfat and y gallons of milk containing 2% butterfat are used.
The total amount of milk is x + y gallons, and we want it to be equal to 258 gallons.
To determine the amount of butterfat in the mixture, we can multiply the volume of each type of milk by its respective butterfat percentage and sum them up.
For milk containing 8% butterfat, the amount of butterfat is 0.08x (8% is equivalent to 0.08 as a decimal).
For milk containing 2% butterfat, the amount of butterfat is 0.02y (2% is equivalent to 0.02 as a decimal).
Since we want the final mixture to contain 7% butterfat, we can set up the following equation:
0.08x + 0.02y = 0.07(258)
Simplifying the equation, we have:
0.08x + 0.02y = 18.06
To solve for x and y, we need another equation. Since the total amount of milk is x + y = 258, we can rearrange it to y = 258 - x.
Substituting this value into the equation above, we get:
0.08x + 0.02(258 - x) = 18.06
Solving this equation will give us the values of x and y, which represent the gallons of milk containing 8% butterfat and 2% butterfat, respectively, needed to obtain the desired 258 gallons.
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A population of rare birds in town is currently listed at 2,000. It is declining at a rate of 2% per year. How many birds will be left after 20 years? Round your answer to the nearest whole number.
A. 1,335 birds
B. 1,980 birds
C. 2,972 birds
D. 23 birds
Option(A) is the correct answer is A. 1,335 birds.
To calculate the number of birds that will be left after 20 years, we need to consider the annual decline rate of 2%.
We can use the formula for exponential decay:
N = N₀ * (1 - r/100)^t
Where:
N is the final number of birds after t years
N₀ is the initial number of birds (2,000 in this case)
r is the annual decline rate (2% or 0.02)
t is the number of years (20 in this case)
Plugging in the values, we get:
N = 2,000 * (1 - 0.02)^20
N = 2,000 * (0.98)^20
N ≈ 2,000 * 0.672749
N ≈ 1,345.498
Rounded to the nearest whole number, the number of birds that will be left after 20 years is 1,345.
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Let Q be an orthogonal matrix with an eigenvalue λ1=1. Let x be an eighenvector beloinging to λ1. Show that x is also an eigenvector of QT
If Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).
To show that x is also an eigenvector of QT, we need to demonstrate that QT * x is a scalar multiple of x.
Given that Q is an orthogonal matrix, we know that QT * Q = I, where I is the identity matrix. This implies that Q * QT = I as well.
Let's denote x as the eigenvector corresponding to the eigenvalue λ1 This means that Q * x = λ1 * x.
Now, let's consider QT * x. We can multiply both sides of the equation Q * x = λ1 * x by QT:
QT * (Q * x) = QT * (λ1 * x)
Applying the associative property of matrix multiplication, we have:
(QT * Q) * x = λ1 * (QT * x)
Using the fact that Q * QT = I, we can simplify further:
I * x = λ1 * (QT * x)
Since I * x equals x, we have:
x = λ1 * (QT * x)
Now, notice that λ1 * (QT * x) is a scalar multiple of x, where the scalar is λ1. Therefore, we can rewrite the equation as:
x = λ2 * x
where λ2 = λ1 * (QT * x).
This shows that x is indeed an eigenvector of QT, with the eigenvalue λ2 = λ1 * (QT * x).
In conclusion, if Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).
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NO LINKS!! URGENT HELP PLEASE!!!
12. y = -2(x + 3)^3 + 1
1a. What type of function?
2b. How did you translate the parent function to produce the equation?
Answer:
polynomial; left 3 right 1
Step-by-step explanation:
The type of function is a polynomial.
The parent function is translated by moving -2x^3 to the left 3 and right 1.
Answer:
The function is a quadratic function.
Series of transformations:
Horizontal translation of 3 units left.Vertical stretch by a factor of 2.Reflection in the x-axis.Vertical translation of 1 unit up.Step-by-step explanation:
A quadratic function is a polynomial function of degree 2 (the highest exponent of the variable is 2).
Therefore, the given equation, y = -2(x + 3)² + 1, represents a quadratic function as the term (x + 3)² indicates that the highest power of the x-variable is 2.
[tex]\hrulefill[/tex]
The parent function of a quadratic function is y = x².
To translate the parent function to produce the given equation, we need to apply this series of transformations:
1. Horizontal translation
When "a" is added to the x-variable, the graph is shifted "a" units to the left. Therefore, the addition of 3 inside the parentheses shifts the graph horizontally to the left by 3 units.
2. Vertical Stretch
The coefficient in front of the squared term (x + 3)² indicates a vertical stretch or compression. In this case, since the coefficient greater than 1, it stretches the graph vertically, making it narrower compared to the parent function.
3. Reflection
As the stretch coefficient is negative, the graph is reflected across the x-axis, meaning the parabola opens downwards.
4. Vertical translation
When "a" is added to the function, the graph is shifted "a" units up. Therefore, the addition of 1 to the function shifts the graph vertically up by 1 unit.
In summary, the series of transformations that maps the parent function to the given function is:
Horizontal translation of 3 units left.Vertical stretch by a factor of 2.Reflection in the x-axis.Vertical translation of 1 unit up.Sample response: The product of two numbers with
different signs is negative, so 2(-12) = -24, not 24. Then
-24-(-30) = -24 + 30 = 6.
Select all the information you considered when writing
your response.
The product or quotient of two integers with
different signs is negative.
To subtract an integer, add its opposite.
To add integers with opposite signs, subtract the
absolute values. The sum has the same sign as the
integer with the greater absolute value.
By considering these rules and properties of integers, the correct result of 6 was obtained.
When writing the response, I considered the following information:
The product or quotient of two integers with different signs is negative. This rule was used to determine that 2(-12) equals -24, not 24.
To subtract an integer, add its opposite. This rule was applied when subtracting -30 from -24, resulting in -24 - (-30) = -24 + 30.
To add integers with opposite signs, subtract the absolute values. The sum has the same sign as the integer with the greater absolute value.
This rule was used to calculate -24 + 30 = 6, where the absolute value of 30 is greater than the absolute value of -24.
By considering these rules and properties of integers, the correct result of 6 was obtained.
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The roster method of the set A' = {14, 15, 20}
How to find sets using roster method?The roster method is defined as a way to show the elements of a set by listing the elements inside of brackets.
A typical example of the roster method is to write the set of numbers from 1 to 5 as {1, 2, 3, 4, 5}.
Therefore,
U = universal sets = {14, 15, 16, 17, 18, 19, 20}
A = {16, 17, 18, 19}
Therefore, let's find A' as follows:
A' is the value not in A. Hence, in roster method.
A' = {14, 15, 20}
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10 marbles 6 black 4 white what is the decimal probability of getting a white
Answer:
.6
Step-by-step explanation:
Two sides of a triangle have lengths 43 and 67. The angle included between these sides measures 27degrees°. To the nearest hundreth, what is the length of the third side?
To the nearest hundredth, the third side of the triangle is 62.74.
To find the length of the third side of the triangle, we can use the Law of Cosines, which relates the lengths of the sides and the cosine of an angle in a triangle. The formula is as follows:
[tex]c^2[/tex] = [tex]a^2[/tex] + [tex]b^2[/tex] - 2ab*cos(C)
Where:
c = length of the third side,
a and b = lengths of the other two sides, and
C = included angle.
Given that the lengths of the two sides are 43 and 67, and the included angle is 27 degrees, we can substitute these values into the formula:
[tex]c^2[/tex] = [tex]43^2[/tex] + [tex]67^2[/tex] - 2 * 43 * 67 * cos(27°)
Now we can calculate the value of c:
[tex]c^2[/tex] ≈ 1849 + 4489 - 2 * 43 * 67 * 0.8910065242
[tex]c^2[/tex] ≈ 1849 + 4489 - 5946.654213618
[tex]c^2[/tex] ≈ 3931.345786382
Taking the square root of both sides to isolate c, we get:
c ≈ √3931.345786382
c ≈ 62.74
Therefore, to the nearest hundredth, the length of the third side of the triangle is approximately 62.74.
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In 2020 there were 18,700 students at college a with a projected enrollment increase of 500 students per year in the same year there were 26,200 students at college B with a projected enrollment decline of 1000 students per year according to these projections when will the colleges have the same enrollment? What will be the enrollment in each college at that time?
In 2025, both colleges will have an enrollment of 21,200 students.
To determine when the colleges will have the same enrollment, we need to set up an equation based on the projected enrollment increases and declines.
Let's assume the number of years from 2020 is represented by 'x'.
For College A, the projected enrollment can be represented by: 18,700 + 500x.
For College B, the projected enrollment can be represented by: 26,200 - 1000x.
To find when the colleges will have the same enrollment, we need to solve the equation:
18,700 + 500x = 26,200 - 1000x
Combining like terms, we get:
1500x = 7,500
Dividing both sides by 1500, we find:
x = 5
Therefore, the colleges will have the same enrollment in 5 years, or in the year 2025.
To determine the enrollment at that time, we substitute x = 5 into either equation. Let's use College A's equation:
Enrollment in College A in 2025:
18,700 + 500(5) = 18,700 + 2500 = 21,200 students
Enrollment in College B in 2025:
26,200 - 1000(5) = 26,200 - 5000 = 21,200 students
Both colleges will have an enrollment of 21,200 students in the year 2025 when they have the same enrollment.
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What is the meaning of "Then S ∖m [tex]\neq[/tex]∅; otherwise S is a subset of a finite set; a contradiction"?
The phrase "Then S ∖m ∅; otherwise S is a subset of a finite set; a contradiction" is expressing a logical statement and its implications. Let's break it down:
"S ∖m ∅" means that the set difference between S and m is not empty. In other words, there is at least one element in S that is not in m."Otherwise S is a subset of a finite set" suggests that if the set difference is empty, then S must be a subset of a finite set. This implies that S has a limited number of elements.A contradiction" indicates that the statements in points 1 and 2 cannot both be true simultaneously because they contradict each other. If S has at least one element not in m (point 1), it cannot be a subset of a finite set (point 2).In summary, the statement is asserting that if the set difference between S and m is not empty, it contradicts the notion that S is a subset of a finite set.For such more question on element
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Decide whether the rates are equivalent. Maria saves $50 in 4 months.
Ralph saves $60 in 5 months
Answer:
The rates are not equivalent since Maria saves $0.50 more per month than Ralph.
Step-by-step explanation:
We can determine if two rates are equivalent by comparing the rates at which they save per month.
Maria's savings per month:
Both 50 and 4 can be divided by 2, which gives us 25/2. As a regular number, this becomes 12.5/1 which means Maria saves $12.5 per month.
Ralph's savings per month:
Both 60 and 5 can be divided by 5, which gives us 12. Thus, Ralph saves $12 per month.
Thus, the rates are not equivalent as Maria saves $0.50 more per month than Ralph.
In a popular online role playing game, players can create detailed designs for their character's "costumes," or appearance. Tariq sets up a website where players can buy and sell these costumes online. Information about the number of people who visited the website and the number of costumes purchased in a single day is listed below.
The probability that a person will purchase no more than one costume is given as follows:
93%.
How to calculate a probability?The parameters that are needed to calculate a probability are listed as follows:
Number of desired outcomes in the context of a problem or experiment.Number of total outcomes in the context of a problem or experiment.Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.
The total number of outcomes for this problem is given as follows:
187 + 228 + 29 = 444.
Only 29 people purchased more than one costume, hence the probability of at most one costume is calculated as follows:
(444 - 29)/444 = 0.93 = 93%.
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the equation below represents the total price of Michigan State University per semester where c represents the number of classes and t represents the total cost for the semester including a one time fee for room and board
The equation that represents the total price of Michigan State University per semester is determined by the number of classes taken (c) and the total cost for the semester, which includes a one-time fee for room and board (t).
The equation captures the relationship between these variables and calculates the overall cost for a student attending the university. It is important to note that the equation provides a quantitative representation and does not take into account other factors such as scholarships, financial aid, or additional expenses. Therefore, the equation serves as a useful tool for estimating the total cost of attending Michigan State University, allowing students and their families to plan and budget accordingly for their education.For such more question on equation
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100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answer:
Step-by-step explanation:
According to the triangle proportionality theorem (or thales theorem), the lines are parallel if AD/BD = AE/CE
15/12= 1.25
10/8 = 1.25
Because the ratios are equal, the lines are parallel to each other.
The come shown in the diagram has a circular base with a radius of 6 inches perpendicular to the height. The cone is 414.7 cubic inches. What is the height,h, of the cone to the nearest whole inch? If the length of the radius is doubled and the height of the cone changed to 8 inches, find the volume of the new cone.
The volume of the new cone is 1,536π cubic inches.
To find the height of the cone, we can use the formula for the volume of a cone:
V = (1/3)πr²h
Given that the volume of the cone is 414.7 cubic inches and the radius is 6 inches, we can substitute these values into the formula and solve for h:
414.7 = (1/3)π(6)²h
414.7 = (1/3)π(36)h
414.7 = 12πh
h = 414.7 / (12π)
h ≈ 10.93 inches
Therefore, the height of the cone is approximately 10.93 inches when rounded to the nearest whole inch.
Next, let's calculate the volume of the new cone after doubling the radius to 12 inches and changing the height to 8 inches.
Using the same formula, we have:
V = (1/3)π(12)²(8)
V = (1/3)π(144)(8)
V = 1,536π
The redesigned cone has 1,536 cubic inches of capacity.
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The equation below represents the total price of Michigan State University per
semester, where c represents the number of classes and T represents the total cost
for the semester, including a one time fee for room and board.
T = 1473c + 5495
What number represents the slope?
Interpret what the slope means in this situation.
What number represents the y-intercept?
Interpret what the y-intercept means in the situation.
The slope is represented by the number 1473 in the equation T = 1473c + 5495. The slope denotes the rate of change, or the amount by which the overall cost (T) rises with each new class attended (c).
In this scenario, the total cost of the semester increases by $1473 for each extra class attended at Michigan State University. The slope reflects the linear relationship between the total cos and the number of classes taken.
The number 5495 represents the y-intercept in the equation. When the number of classes (c) is 0, the y-intercept is the value of T. The y-intercept of $5495 shows the total cost for a semester at Michigan State University when no classes are taken. It includes a one-time accommodation and board price.
The y-intercept can be understood as the university's fixed cost component, which is independent of the number of classes taken. It includes expenditures such as housing and food plans that are incurred regardless of how many classes a student enrolls in.
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The tables of ordered pairs represent some points on the graphs of two lines. What is the solution to the system of equations represented by the two lines? There are no zeroes on the chart. What do I do
To find the solution to the system of equations represented by the two lines based on the given tables of ordered pairs, you can follow these steps:
Examine the tables and identify a pattern or relationship between the x-values and y-values for each line.Determine the slope (rate of change) of each line by calculating the difference in y-values divided by the difference in x-values for any two points on the line.Once you have the slopes, compare them to see if they are equal or different.If the slopes are different, the lines intersect at a single point, which represents the solution to the system of equations.If the slopes are equal, the lines are parallel, and there is no solution to the system of equations.If the slopes are different, you can find the intersection point by solving the system of equations using either substitution, elimination, or another suitable method.For such more question on equations
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Help Seve & Ext
Lusk Corporation produces and sells 15,100 units of Product X each month. The selling price of Product X is $21 per unit, and variable expenses are $15 per unit. A
study has been made concerning whether Product X should be discontinued. The study shows that $72,000 of the $101000 in monthly fixed expenses charged to
Product would not be avoidable even if the product was discontinued if Product X is discontinued, the monthly financial advantage (disadvantage) for the company
Subm
If Product X is discontinued, the company would have a monthly financial advantage of $61,600.
To determine the monthly financial advantage or disadvantage for the company if Product X is discontinued, we need to calculate the total revenue and total expenses associated with Product X.
Total Revenue:
Product X sells 15,100 units per month at a price of $21 per unit. Therefore, the total revenue can be calculated as:
Total Revenue = Number of Units Sold × Price per Unit
Total Revenue = 15,100 units × $21 = $317,100
Total Variable Expenses:
The variable expenses per unit for Product X are given as $15. Therefore, the total variable expenses can be calculated as:
Total Variable Expenses = Number of Units Sold × Variable Expenses per Unit
Total Variable Expenses = 15,100 units × $15 = $226,500
Total Fixed Expenses:
Out of the $101,000 in monthly fixed expenses, $72,000 would still be incurred even if Product X is discontinued. Therefore, the avoidable fixed expenses would be:
Total Fixed Expenses - Avoidable Fixed Expenses = $101,000 - $72,000 = $29,000
Monthly Financial Advantage (Disadvantage):
To calculate the monthly financial advantage or disadvantage, we subtract the total expenses (fixed and variable) from the total revenue:
Monthly Financial Advantage (Disadvantage) = Total Revenue - Total Variable Expenses - Total Fixed Expenses
Monthly Financial Advantage (Disadvantage) = $317,100 - $226,500 - $29,000
Monthly Financial Advantage (Disadvantage) = $61,600
Therefore, if Product X is discontinued, the company would have a monthly financial advantage of $61,600.
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The length of a classroom is 34 feet. What is this measurement in yards and feet?
Answer:
11 yards and 3 feet
Please awnser asap I will brainlist
Answer: true
Step-by-step explanation:
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Answer:
a) ∩
Step-by-step explanation:
set A : {6,8,10,12}
set B : {5,6,7,8,9}
set C : {6,8}
A ∪ B = {6,8,10,12} ∪ {5,6,7,8,9} = {5,6,7,8,9,10,12} ≠ C
A ∩ B = {6,8,10,12} ∩ {5,6,7,8,9} = {6, 8} = C
Therfore{6,8,10,12} ∩ {5,6,7,8,9} = {6, 8}
Identify the number of solutions of the system of linear equations.
x=y+3z=6
x-2y = 5
2x - 2y + 5z = 9
no solution
exactly one solution
infinitely many solutions
Solve the system. If there are infinitely many solutions, write the ordered triple in terms of z. If there is no solution, lea
The solution is (x, y, z)
-1.2
3
The solution to the system of linear equations is (x, y, z) = (64/11, 37/11, 9/11). The system has exactly one solution.
Let's solve the system of linear equations correctly.
The given system of linear equations is:
x = y + 3z = 6
x - 2y = 5
2x - 2y + 5z = 9
To determine the number of solutions, we can analyze the system using the method of elimination or substitution. Let's use the method of elimination:
From equation 1, we have:
x = y + 3z
Substituting this value of x in equation 2:
(y + 3z) - 2y = 5
y + 3z - 2y = 5
-z + 3z = 5 - y
2z = 5 - y
Now, let's substitute the value of x in equation 3:
2(y + 3z) - 2y + 5z = 9
2y + 6z - 2y + 5z = 9
11z = 9
Simplifying the equation, we find:
z = 9/11
Now, substituting this value of z back into the equation 2z = 5 - y, we get:
2(9/11) = 5 - y
18/11 = 5 - y
18/11 - 5 = -y
18/11 - 55/11 = -y
-37/11 = -y
y = 37/11
Finally, we can substitute the values of y and z into equation 1 to find the value of x:
x = y + 3z
x = 37/11 + 3(9/11)
x = 37/11 + 27/11
x = 64/11
Therefore, the solution to the system of linear equations is (x, y, z) = (64/11, 37/11, 9/11).
The system has exactly one solution.
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HELP this doesn’t have anything instructions and I’m confused on what do to
A. 25 = 27
B. 21+22= 180°
C. 24 28
D.26 +27= 180°
The angles formed by the transversal line r and the parallel lines s and t, have that:
A. ∠5 ≈ ∠7 {vertical angles}
B. ∠1 + ∠2 = 180° {angles on straight line}
C. ∠4 ≈ ∠8 {corresponding angles}
D. ∠6 + ∠7 = 180° {angles on straight line}
What are angles formed by a pair of parallel lines cut by a transversal line?When a transversal line intersects a pair of parallel lines, several angles are formed which includes: Corresponding angles, vertical angles, and alternate angles.
A. The angles ∠5 and ∠7 are vertical angles as they are vertically opposite, so ∠5 = ∠7
B. The angles ∠1 and ∠2 angles on a straight line and their sum is equal to 180° so;
∠1 + ∠2 = 180°
C. The angles ∠4 and ∠8 are corresponding angles and are equal, so ∠4 = ∠8
D. The angles ∠6 and ∠7 angles on a straight line and their sum is equal to 180° so;
∠6 + ∠7 = 180°
Therefore, for the angles formed by the transversal line r and the parallel lines s and t, we have that:
A. ∠5 ≈ ∠7 {vertical angles}
B. ∠1 + ∠2 = 180° {angles on straight line}
C. ∠4 ≈ ∠8 {corresponding angles}
D. ∠6 + ∠7 = 180° {angles on straight line}
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O 2. Draw two more arrangements with dots to form the sequence: 1; 3; 6; 10; 15;... ล th th 3. How many dots will there be in the 9 and 10" arrangements? Explain how you got your answer. (3) 4. What do you observe if you take any two consecutive arrangements and add the number of dots? (2) [19]
The 9th arrangement will have 36 dots, the 10th arrangement will have 45 dots, and when adding the number of dots in any two consecutive arrangements, the result will be the number of dots in the next arrangement.
To determine the number of dots in the 9th and 10th arrangements, we need to understand the pattern in the given sequence: 1, 3, 6, 10, 15, ...
To find the pattern, we can observe the differences between consecutive terms: 2, 3, 4, 5, ...
We notice that these differences increase by 1 each time. This indicates that the sequence is formed by adding consecutive positive integers to the previous term.
Using this pattern, we can determine the 9th and 10th arrangements as follows:
9th arrangement:
Starting with the 1st arrangement, we add the positive integers consecutively: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.
Therefore, the 9th arrangement will have 36 dots.
10th arrangement:
Starting with the 1st arrangement, we add the positive integers consecutively: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
Therefore, the 10th arrangement will have 45 dots.
Now, let's address the observation when we take any two consecutive arrangements and add the number of dots.
If we take the nth arrangement and the (n+1)th arrangement, we can observe that the number of dots in the (n+1)th arrangement will be the sum of the number of dots in the nth arrangement and the number of dots added in the (n+1)th step.
For example, let's consider the 2nd and 3rd arrangements:
2nd arrangement: 3 dots
3rd arrangement: 3 + 3 (adding 2, the next positive integer) = 6 dots
This pattern holds true for any two consecutive arrangements in the sequence.
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The number of subsets for the set in this problem is given as follows:
16 subsets.
How to obtain the number of subsets in a set?Considering a set with n elements, the number of subsets in the set is the nth power of 2, that is:
[tex]2^n[/tex]
The set in this problem is given as follows:
{C, D, E, F}.
Hence the cardinality of the set is given as follows:
n = 4.
Then the number of subsets is given as follows:
[tex]2^4 = 16[/tex]
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