Which of the following is an equation of the line through (2, 3) and (-1, -12)?

Answer:

x=0

Step-by-step explanation:

-14-8x=-2(-3x+7)

-14-8x=6x-14

-8x=6x+0

-2x=0

x=0

For every matrix A in the space of linear transformations from [tex]R^n[/tex] to [tex]R^1[/tex], there exists a unique vector y such that Ax = x.y

The norm of A is equal to the absolute value of y. This is proven using the Schwarz inequality.

Let A be a matrix in the space of linear transformations from [tex]R^n[/tex]  to [tex]R^1[/tex]. For any vector x in [tex]R^n[/tex], we can define Ax as a dot product between A and x. Let y be a vector in [tex]R^1[/tex] such that Ax = x.y for all x in [tex]R^n[/tex] . This means that the dot product of A and x yields a scalar multiple of x, where the scalar is y.

To prove uniqueness, assume there are two vectors [tex]y_{1}[/tex] and [tex]y_{1}[/tex] in [tex]R^1[/tex] such that Ax = x.[tex]y_{1}[/tex] and Ax = x.[tex]y_{2}[/tex] for all x in [tex]R^n[/tex]. Then, x.[tex]y_{1}[/tex] = x.[tex]y_{2}[/tex] for all x in [tex]R^n[/tex], which implies [tex]y_{1}= y_{2}[/tex]. Therefore, there exists a unique vector y such that Ax = x.y.

To prove that ||A|| = |y|, we can use the Schwarz inequality. The Schwarz inequality states that for any vectors u and v in [tex]R^n[/tex], |u.v| ≤ ||u|| * ||v||, with equality holding if and only if u and v are linearly dependent. Applying this inequality to the vectors A and x, we have |Ax.x| ≤ ||Ax|| * ||x||. Since Ax = x.y and ||Ax|| = ||x.y|| = |y| * ||x||, the inequality becomes |x.y.x| ≤ |y| * [tex]||x||^2[/tex]. Since the dot product x.y.x is equal to y * (x.x) = y * [tex]||x||^2[/tex], we have |y| *[tex]||x||^2[/tex] ≤ |y| * [tex]||x||^2[/tex], which holds with equality. Hence, ||A|| = |y|.

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Our goal is to find values of X, Y₁, and Y₂ that make both equations equal to zero.The unknown Variables: X, Y₁, Y₂,The equations: Equation (1) and Equation (2)

In the given equations, we have three unknown variables: X, Y₁, and Y₂. These variables represent the values we are trying to find.

We also have two equations, labeled as Equation (1) and Equation (2). These equations provide the relationships between the variables.

To solve for the unknown variables, we need to find values of X, Y₁, and Y₂ that satisfy both equations simultaneously.

Let's restate the equations for clarity:

Equation (1): X - a + bY₁ = 0

Equation (2): X - e - dY₂ = 0

Our goal is to find values of X, Y₁, and Y₂ that make both equations equal to zero.

So, in summary:

- The unknown variables: X, Y₁, Y₂

- The equations: Equation (1) and Equation (2)

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Answer:

3 is the coefficient

Step-by-step explanation:

3x + 2 = -10

The coefficient is the number that is multiplied by the variable

3 is the coefficient

Answer:

3

Step-by-step explanation:

the 'a' coefficient is 3

3 is the number infront of x (the first one)

a coefficient is the number infront of the unknown value

10x+4x-1=0

10= coefficient a

4= coefficient b

-1= coefficient c

Given :

Two companies offer jane sales position. Company a pays $3,000 per month plus $250 per car sold.. Company B pays 2,500 per month plus $350 per car sold.

To Find :

Which equation could be used to determine the number of cars sold in a month, x, that would result in the same total income from each company.

Solution :

Let, x car are sold for same income.

Mathematical equation of company one :

[tex]T_1 =3000+250x[/tex]

Mathematical equation of company two :

[tex]T_2 =2500+350x[/tex]

For same income :

[tex]T_1=T_2\\\\3000 + 250x = 2500 + 350x\\\\100x = 500\\\\x=5[/tex]

So, for same total income from each company the number of cars sold in a month should be 5.

Hence, this is the required solution.

The probability that a randomly selected day in December will have an average temperature of at least 60°F is approximately 0.1935 (or 19.35%).

To calculate the probability of randomly selecting a day with an average temperature of at least 60°F, you need additional information. Specifically, you need to know the total number of days in December in Orange County, CA.

Since you mentioned that there are 31 days in the month of December, we'll assume that this is accurate. In that case, the probability can be calculated based on the 31-day period.

Given that the dataset consists of 20 days and only 6 of them have an average temperature of at least 60°F, you can calculate the probability as follows:

Probability = (Number of days with average temperature ≥ 60°F) / (Total number of days in December)

Probability = 6 / 31

So, the probability that a randomly selected day in December will have an average temperature of at least 60°F is approximately 0.1935 (or 19.35%).

Please note that if the actual number of days in December differs from 31 or if you have access to a larger dataset, the probability calculation would change accordingly.

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Answer:

46 laps

Step-by-step explanation:

Set up a ratio:

38/95 =  s/115

cross multiply

4370  = 95s

divide both sides by 95

s= 46

Answer:

1) x=67, y=83, z=80 2) x=5

Step-by-step explanation:

1) 28+85=113 a triangle is 180 degrees so 180-113=67 so x=67

x and y are supplementary meaning both sides equal 150 so 150-67=83

83+17=100 now going back how much did i say a triangle should equal? 180 so z is 80

2)a triangle = 180 since the triangle is a right angle it includes the angle 90 so make the equation to solve for x

180=90+8x+2+9x+3

take 90 to other side so you are subtracting 180-90

90=8x+2+9x+3

combine like terms

90=17x+5

subtrcat 5 from 90

85=17x

now divide 85 by 17 and you get x=5

To land the drone on the roof of the building, it would need to fly approximately 1505.35 feet in a straight line. The angle between the ground and the drone's flight path would be approximately 5.74 degrees.

To determine the distance the drone needs to fly, we can use the Pythagorean theorem. The horizontal distance between your current position in the park and the building is given as 1500 feet, and the height of the building is 150 feet. We can consider this as a right-angled triangle, with the horizontal distance as the base and the height of the building as the perpendicular side.

Using the Pythagorean theorem, we can find the hypotenuse (flight distance) of the triangle, which represents the straight-line distance the drone needs to travel. The equation is:

hypotenuse^2 = base^2 + perpendicular^2

Substituting the values, we get:

hypotenuse^2 = 1500^2 + 150^2

hypotenuse^2 = 2250000 + 22500

hypotenuse^2 = 2272500

Taking the square root of both sides, we find:

hypotenuse ≈ √2272500 ≈ 1505.35 feet

Therefore, the drone would need to fly approximately 1505.35 feet in a straight line to reach the roof of your building.

To determine the angle of the flight path, we can use trigonometry. The angle (θ) can be calculated using the tangent function:

tan(θ) = perpendicular/base

Substituting the values, we get:

tan(θ) = 150/1500

tan(θ) = 0.1

Taking the inverse tangent (arctan) of both sides, we find:

θ ≈ arctan(0.1) ≈ 5.74 degrees

Hence, the drone should take an angle of approximately 5.74 degrees in its flight path to reach the roof of your building.

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Answer:

x=-3.5

Step-by-step explanation:

-3x=10.5 divide both sides by -3

Answer:

x = -3.5

Step-by-step explanation:

-3x - 4.6 = 5.9 ( add 4.6 on both sides)

-3x = 10.5 (divide by -3 on both sides)

x = -3.5

(a) The matrix polynomial P(x) = x² can be evaluated by substituting the given 2 by 2 matrix A = [-1, 0; -2, 1] into P(x).

(b) To calculate A^-, where k is any integer, we need to find the inverse of the matrix A. Additionally, P(A) can be obtained by substituting matrix A into the matrix polynomial P(x) = x².

(a) To evaluate the matrix polynomial P(x) = x² using the given 2 by 2 matrix A = [-1, 0; -2, 1], we substitute A into P(x):

P(A) = A² = [-1, 0; -2, 1]² = [(-1)(-1) + (0)(-2), (-1)(0) + (0)(1); (-2)(-1) + (1)(-2), (-2)(0) + (1)(1)] = [1, 0; 0, -3].

(b) To calculate A^-, where k is any integer, we need to find the inverse of the matrix A = [2, -2; -2, 4]. The inverse of a 2 by 2 matrix can be found using the formula:

A^-1 = (1/det(A)) * adj(A),

where det(A) is the determinant of A and adj(A) is the adjugate of A. Calculating the determinant and adjugate of A, we have:

det(A) = (2)(4) - (-2)(-2) = 12 - 4 = 8,

adj(A) = [4, 2; 2, 2].

Therefore, A^-1 = (1/8) * [4, 2; 2, 2] = [1/2, 1/4; 1/4, 1/4].

To find P(A), we substitute A into the matrix polynomial P(x) = x²:

P(A) = A² = [2, -2; -2, 4]² = [2, -2; -2, 4] * [2, -2; -2, 4] = [4 - 4, -4 + 8; -4 + 4, 4 + 16] = [0, 4; 0, 20].

Therefore, A^-1 = [1/2, 1/4; 1/4, 1/4] and P(A) = [0, 4; 0, 20].

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Answer:

[tex]\frac{1}{6}[/tex]

Step-by-step explanation:

[tex]\frac{-36}{-216} \\=\frac{36}{216} \\=\frac{1}{6}[/tex]

Answer:

It can divide equally  

Step-by-step explanation:

==============================================

Explanation:

The term "equidistant" means "equally distant".

We want (x,y) to be the same distance to the x axis, the y axis, and the point (3,6).

Let n represent this unknown equal distance. Negative distance doesn't make sense, so we'll have n > 0.

Start at the origin and move n units to the right to arrive at (n,0). We are n units from the y axis.

Now move n units up to arrive at (n,n). This point is guaranteed to be n units away from each axis.

That is one potential point. Other potential points are:

One point for each quadrant.

Let's assume that the answer is in the 1st quadrant since (3,6) is located here.

We need to find an expression that represents the distance from the point (n,n) to (3,6). We'll set that distance expression equal to n so we can solve.

Part 1 of those steps

[tex]d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\\\\n = \sqrt{(n-3)^2+(n-6)^2}\\\\n^2 = (n-3)^2+(n-6)^2\\\\n^2 = (n^2-6n+9)+(n^2-12n+36)\\\\n^2 = 2n^2-18n+45\\\\[/tex]

Part 2 of those steps

[tex]0 = 2n^2-18n+45-n^2\\\\0 = n^2-18n+45\\\\n^2-18n+45 = 0\\\\(n-3)(n-15) = 0\\\\n-3 = 0\ \text{ or } \ n-15 = 0\\\\n = 3\ \text{ or } \ n = 15\\\\[/tex]

The quadratic formula could be used as an alternative pathway.

Therefore, the mystery point could be located at (3,3) or at (15,15)

Use the distance formula to verify each answer. I'll let you do this part.

---------

Extra info:

If you were to perform similar steps with the point (-n,n), then you'll find that no real numbered solutions exist. The two solutions would be complex numbers.

The cases (n,-n) and (-n,-n) have no solutions at all.

Therefore, we have confirmed only case (n,n) would work.

Option 2 has a lower NPV and thus should be chosen as it results in less total payment and better savings. Hence, you would choose option 2 as the Net Present Value of choosing option 2 is -9,903,615.48.

Given:

Purchase price = 12.5 million baht

Down payment = 15%30-year mortgage

We have to calculate the option which is best for us.

Option 1: Mortgage rate = 7.25%

Option 2:Mortgage rate = 6.75%

(a) Monthly Payment of Option 1:

Loan Amount = 12.5 * (100-15)%

= 10,625,000 baht

n = 30 * 12

= 360

i = 7.25%/12

= 0.006042857

Yearly Interest Paid = i * 10,625,000

= 64,515.48 baht

Monthly Interest Paid = 64,515.48 / 12

= 5,376.29 baht

EMI =[tex][P x R x (1+R)^n] / [(1+R)^n-1][/tex]

where P = Loan Amount, R= Interest Rate per month, n= total number of payments.

EMI = [tex][10,625,000 x 0.006042857 x (1+0.006042857)^360] / [(1+0.006042857)^360-1][/tex]

EMI = 69,677.31 baht

(b) Monthly Payment of Option 2:

Loan Amount = 12.5 * (100-15)%

= 10,625,000 baht

n = 30 * 12 = 360

i = 6.75%/12 = 0.005625

Yearly Interest Paid = i * 10,625,000

= 59,765.63 baht

Monthly Interest Paid = 59,765.63 / 12

= 4,980.47 baht

Point cost = 4% of Loan Amount

= 10,625,000 x 4/100

= 425,000 baht

Loan Amount after Deducting Points = 10,625,000 - 425,000

= 10,200,000 baht

EMI = [tex][P x R x (1+R)^n] / [(1+R)^n-1][/tex]

where P = Loan Amount, R= Interest Rate per month, n= total number of payments.

EMI =[tex][10,200,000 x 0.005625 x (1+0.005625)^360] / [(1+0.005625)^360-1][/tex]

EMI = 66,128.92 baht

(c) We have to calculate the NPV of the payments to decide which option is better.

We use the formula to calculate NPV:-

NPV = - [tex][B + (PMT / i) * (1 - (1 + i)^-n)] / ((1 + i)^t)[/tex]

Where B = amount borrowed, PMT = monthly payment, i = interest rate, n = total number of payments, t = year.

NPV =[tex]- [10625000 + (69677.31 / 0.006042857) * (1 - (1 + 0.006042857)^-360)] / ((1 + 0.006042857)^30)[/tex]

= -9,892,571.85 baht

NPV = [tex]- [10200000 + (66128.92 / 0.005625) * (1 - (1 + 0.005625)^-360)] / ((1 + 0.005625)^30)[/tex]

= -9,903,615.48 baht

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The product of the complex numbers (3 + 4i) and (6 + i) is 14 + 27i, where i² = -1.

To find the product of the given complex numbers: (3 + 4i)(6 + i)There are different methods to solve this multiplication problem, but one common way is to use the FOIL method:

First, Outer, Inner, Last. This method involves multiplying each term in the first complex number by each term in the second complex number.

FOIL method: First: (3)(6) = 18Outer: (3)(i) = 3iInner: (4i)(6) = 24iLast: (4i)(i) = -4

Using this method, we can write the product as: (3 + 4i)(6 + i) = 18 + 3i + 24i - 4i²

The term i² equals -1, so we can substitute -1 for i² and simplify: (3 + 4i)(6 + i) = 18 + 27i - 4= 14 + 27i

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Answer:

4,6,8

Step-by-step explanation:

Let x = 1st even integer

x+2 =2nd even integer

x+4 =3nd even integer

x+ x+2 + x+4 = 18

Combine like terms

3x+6 = 18

Subtract 6 from each side

3x+6-6=18-6

3x= 12

Divide by 3

3x/3 =12/3

x = 4

x+2 = 6

x+4 = 8

During the current year, merchandise was sold for $662,100. Of this amount, $114,000 was cash sales and $548,100 was on account. The cost of goods sold was $423,700. Therefore, the gross profit was $238,400.

The gross profit is calculated by subtracting the cost of goods sold from the total sales. The cost of goods sold is the direct cost of producing the goods that were sold. It includes the cost of materials, labor, and overhead.

The gross profit is an important measure of a company's profitability. It shows how much money the company makes from its core business activities, before deducting other expenses.

In this case, the company's gross profit margin is 35.8%. This means that for every $100 of sales, the company makes $35.80 in gross profit. A high gross profit margin is generally considered to be a sign of a healthy business.

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Given:

Baseball is thrown at 89 miles per hour.

The ball will travel 324 feet when hit by a bat swung at 51 miles per hour.

It will travel 444 feet when hit by a boat swung at 81 miles per hour.

To find:

The addition distance travelled by the ball if the speed of bat increased by 1 unit.

Step-by-step explanation:

Let y be the number of feet traveled by the ball when hit by a bat swung at x miles per hour.  

The ball will travel 324 feet when hit by a bat swung at 51 miles per hour. So, the point is (51,324).

It will travel 444 feet when hit by a boat swung at 81 miles per hour. So, the point is (81, 444).

Equation of line passes through two points is

[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]

[tex]y-324=\dfrac{444-324}{81-51}(x-51)[/tex]

[tex]y-324=\dfrac{120}{30}(x-51)[/tex]

[tex]y-324=4(x-51)[/tex]

[tex]y-324=4x-204[/tex]

Add 324 on both sides.

[tex]y=4x-204+324[/tex]

[tex]y=4x+120[/tex]

Using the slope intercept form,  

[tex]y=mx+b[/tex]

where, m is slope.

The slope of equation [tex]y=4x+120[/tex] is 4. It means, the ball will travel 4 feet farther for each mile-per-hour increase in the speed of the bat.

The probability of rolling an odd number is 1/2, assuming that the number cube is fair.

The sample space is the set of all possible outcomes of an experiment. In this case, the experiment is rolling a number cube once and recording the outcome. Since the number cube has six faces labeled from 1 to 6, the sample space can be represented as:

S = {1, 2, 3, 4, 5, 6}

The event of rolling an odd number consists of the outcomes 1, 3, and 5.

Therefore, the set of all outcomes for the event of rolling an odd number is: {1, 3, 5}

The probability of rolling an odd number can be found by adding the probabilities of the individual outcomes in the event:

{P(odd)} = P(1) + P(3) + P(5) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2

Therefore, the probability of rolling an odd number is 1/2.

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Answer:

The answer to what?

Step-by-step explanation:

Answer:

-3y2-8+5y2-1=

2y2-9

Step-by-step explanation:

Answer:

[tex]\huge{\boxed{\sf{ 2y^{2} - 9 }}}[/tex]

Step-by-step explanation:

[tex]( - 3y^{2} - 8 ) - ( - 5y^{2}+ 1 )[/tex]

While subtracting , sign of each term of the second expression changes.

⇒[tex]- 3y^{2} - 8 + 5y^{2} - 1[/tex]

Combine like terms : [tex]-3y^{2}[/tex] and [tex]5y^{2}[/tex]

Like terms are those which have the same base.

⇒[tex]2y^{2} - 8 - 1[/tex]

Remember : The negative integers are always added but posses the negative ( - ) sign.

⇒[tex]2y^{2} - 9[/tex]

Hope I helped!

Best regards !

-[tex]\sf{ TheAnimeGirl}[/tex]

Answer:

9997.46

Step by step Explanation

Answer:

11 music lessons.

Step-by-step explanation:

We know that membership costs $165 and members pay $25 per music lesson.

So, we can write the following expression:

[tex]165+25m[/tex]

The 165 represents the one-time membership fee and the 25m represents the cost for m music lessons.

We know that non-members pay no membership fee but their cost per lesson is $40. So:

[tex]40m[/tex]

Represents the cost for non-members for m music lessons.

We want to find how many music lessons would have to be taken for the cost to be the same for both members and non-members. So, we can set the expressions equal to each other:

[tex]165+25m=40m[/tex]

And solve for m. Let's subtract 25m from both sides:

[tex]165=15m[/tex]

Now, divide both sides by 15:

[tex]m=11[/tex]

So, at the 11th music lesson, members and non-members will pay the same.

Further Notes:

This means that if a person would only like to take 10 or less lessons, the non-membership is best because there is no initial fee.

However, if a person would like to take 12 or more lessons, than the membership is best because the membership has a lower cost per lesson than the non-membership.

And we're done!

Answer:

5.  This question is hard to explain, sorry.

The interquartile range (IQR) of the data set is 7.To find the interquartile range (IQR) of a data set, we need to first determine the values of the first and third quartiles.

The first quartile (Q1) represents the 25th percentile of the data, while the third quartile (Q3) represents the 75th percentile of the data. We can find these values by first sorting the data set in ascending order:

0, 0, 0, 1, 2, 2, 4, 5, 7, 8, 10

Next, we can find the median (Q2) of the entire data set, which is the value that separates the lower 50% of the data from the upper 50%:

0, 0, 0, 1, 2, | 2, 4, 5, 7, 8, 10

Q2 = 2

Now we can find Q1 and Q3. To find Q1, we take the median of the lower half of the data set:

0, 0, 0, | 1, 2,

Q1 = 0

To find Q3, we take the median of the upper half of the data set:

2, 4, 5, 7, 8, 10

Q3 = 7

Finally, we can calculate the interquartile range (IQR) as the difference between Q3 and Q1:

IQR = Q3 - Q1

= 7 - 0

= 7

Therefore, the interquartile range (IQR) of the data set is 7.

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Answer: 7

Step-by-step explanation:

Answer: x ≥ 5

---------------------

Answer:

Step-by-step explanation:

Try using g(x)

Let g(x) = y

y = 2x + 4

Interchange x and y

x = 2y + 4

Subtract 4 from both sides

x - 4 = 2y

Divide by 2

x/2 - 4/2 = 2y/2

y = x/2 - 2

This is exactly what f(x) is

So g(x) and f(x) are inverses.

Answer:

x² + 2hx + h² - 4

Step-by-step explanation:

To evaluate f(x + h), substitute x = x + h into f(x), that is

fx + h)

= (x + h)² - 4 ← expand (x + h)² using FOIL

= x² + 2hx + h² - 4