a triangular window can only have an area of 120 square feet and the architect wants the base to be 4 feet more than twice the height. find the base and height of the window.

(L2) An inscribed circle is one that encompasses a polygon so that it passes by each of the polygon's vertices.

A circumcircle, not an inscribed circle, is a circle that encircles a polygon at each vertex. A circle that is enclosed within a polygon and intersects each side of the polygon exactly once is said to be inscribed. A circumcircle, on the other hand, is a circle that goes through every vertex of the polygon, with its center located at the point where the perpendicular bisectors of the polygon's sides converge. The greatest circle that can be drawn within a polygon is the circumcircle, while the largest circle that can be drawn inside a triangle is the inscribed circle.

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A) The amount of material used to make the box, A, is equal to 8x^2 + 40/x, where x is the length of a side of the square base.

B) For a base measuring 1 foot by 1 foot, A = 48 square feet.

C) For a base measuring 2 feet by 2 feet, A = 56 square feet.

D) The graph of A = A(x) is a quadratic function with a minimum value. The value of x for which A is smallest can be determined by finding the vertex of the quadratic function.

A) To find the amount of material used, A, we need to consider the surface area of the box. The box has six faces, and since it is closed, all faces need to be accounted for. The four vertical faces form a rectangular prism with dimensions x by x by h, where h is the height of the box.

The area of each face is x * h, and since there are four of them, the total area is 4 * x * h = 4xh. The top face is a square with side length x, and the bottom face is also a square with side length x. Therefore, the total surface area of the box is A = 4xh + 2x^2.

Given that the volume of the box is 10 cubic feet, we have the equation x^2 * h = 10. Solving for h, we get h = 10/x^2. Substituting this back into the surface area equation, we have A = 4x(10/x^2) + 2x^2 = 40/x + 2x^2 = 8x^2 + 40/x.

B) For a base measuring 1 foot by 1 foot, we substitute x = 1 into the equation for A: A = 8(1)^2 + 40/1 = 8 + 40 = 48 square feet.

C) For a base measuring 2 feet by 2 feet, we substitute x = 2 into the equation for A: A = 8(2)^2 + 40/2 = 32 + 20 = 52 square feet.

D) To find the value of x for which A is smallest, we need to find the vertex of the quadratic function A(x) = 8x^2 + 40/x. The vertex of a quadratic function of the form ax^2 + bx + c is given by x = -b/2a. In this case, a = 8 and b = 40, so the x-coordinate of the vertex is x = -40/(2*8) = -5/4.

However, since the side length of the square base cannot be negative, we discard this solution.

Therefore, the value of x for which A is smallest is the positive root of the quadratic function, which is x = √(40/8) = √5.

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Step-by-step explanation:

please mark me as brainlest

Given:

To simplify:

Solving we get,

Hence, the answer is,

The possible values for\($r$ are $r = 0$ and $r = \frac{1}{2}$.\)
The answer is that \($|z|$\) can be either 0 or \($\frac{1}{2}$\).

To find the answer, we need to solve for \($z$\) using the given information.

Let's start by setting up the equations. We know that \($P$\) is represented by\($z$\\$Q$\)

is represented by \($(1+i)z$\), and \($R$\) is represented by

\($2\overline{z}$\)

Since\($|z| < 1$\) we can write \($z$ \\as $z = re ^ {i\theta}$\)

where\($r<1$\) and \($\theta$\) is the argument of \($z$\)

Now let's substitute the representations o\(f $P$, $Q$, and $R$ using $z$:\)

\($P = z = re^{i\theta}$\)

\($Q = (1+i)z = (1+i)re^{i\theta}$\)

\($R = 2\overline{z} = 2re^{-i\theta}$\)

Since \($Q$\) is the conjugate of \($R$\), we can equate the real and imaginary parts of \($Q$\) and \($R$\):

\($Re(Q) = Re(R)$ and $Im(Q) = -Im(R)$\)

Expanding these equations gives us:

\($r\cos\theta = 2r\cos\theta$ (equation 1)\)

\($r\sin\theta = -2r\sin\theta$ (equation 2)\)

From equation 1, we have \($r\cos\theta = 2r\cos\theta$\), which simplifies to \($r = 0$ or $r = \frac{1}{2}$\).

From equation 2, we have \($r\sin\theta = -2r\sin\theta$, which simplifies to $0 = 0$.\)

Since the second equation is always true, it doesn't give us any additional information.

Therefore, the possible values for\($r$ are $r = 0$ and $r = \frac{1}{2}$.\)
The main answer is that \($|z|$\) can be either 0 or \($\frac{1}{2}$\).

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The points $P,$ $Q,$ and $R$ are represented by the complex numbers $z,$ $(1 i) z,$ and $2 \overline{z},$ respectively, where $|z|

Answer:

4√mn/m^2

Step-by-step explanation:

√16n/m^3

= √16n/√m^3

= √4x4xn/√mxmxm

= 4√n/m√m

Rationalize by multiplying the numerator and the denominator by the denominator, which is a surd:

= (4√n x √m)/(m√m x √m)

= 4√mxn/m√mxm

= 4√mn/mxm

= 4√mn/m^2

A flexible budget is used to compare the budgeted costs and actual costs in a given period. It allows for adjustments based on the actual level of production or activity, providing a more accurate assessment of cost performance.

A flexible budget is a budgeting tool that adjusts the planned budget based on the actual level of activity or production achieved. It enables a comparison between the budgeted costs and the actual costs incurred during a period.

In the given scenario, Lannister Inc. initially planned to produce 198,000 units and estimated 99,000 direct labor hours. However, the actual production and direct labor hours were 192,000 units and 90,000 hours, respectively.

A flexible budget would compare the budgeted costs with the actual costs based on the actual level of production and direct labor hours. It allows for more accurate cost performance evaluation by incorporating the deviations in activity levels from the initial plan.

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The z-score for P(? ≤ z ≤ ?) = 0.60 is approximately 0.25.

The z-score for P(z ≥ ?) = 0.30 is approximately -0.52.

How to find the Z score

P(Z ≤ z) = 0.60

We can use a standard normal distribution table or a calculator to find that the z-score corresponding to a cumulative probability of 0.60 is approximately 0.25.

Therefore, the z-score for P(? ≤ z ≤ ?) = 0.60 is approximately 0.25.

For the second question:

We want to find the z-score such that the area under the standard normal distribution curve to the right of z is 0.30. In other words:

P(Z ≥ z) = 0.30

Using a standard normal distribution table or calculator, we can find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52 (since we want the area to the right of z, we take the negative of the z-score).

Therefore, the z-score for P(z ≥ ?) = 0.30 is approximately -0.52.

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The rate of change of the function that has a graph that passes

through the points (-1, 3) and (-2,-4) is slope and is equal to 7.

We have given two points  (-1, 3) and (-2,-4) .

Rate of change of graph is given by slope

Using slope formula , we get

      m = -4 - 3 / -2 - (-1)

      m = -7 / -2 + 1

      m = -7 / -1

       m = 7

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Answer:

300%

Step-by-step explanation:

Your technically just multiplying by 3. Hope this helps

Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.

A three-dimensional object's volume is a measurement of how much space it takes up. It is a real-world physical number that can be expressed in cubic measurements like cubic metres (m3), cubic centimetres (cm3), or cubic feet (ft3). Physics, chemistry, architecture, and mathematics all use the idea of volume extensively. Volume is frequently used to refer to the amount of space that an object or substance takes up, for instance the amount of a container, the volume of either a liquid, or the quantity of a gas. Depending on an object's shape, a different formula is required to determine its volume.

given

The formula V = (1/3)r2h, where V is the volume, r is the radius of the base, and h is the height, can be used to determine the volume of a cone.

Hence, by multiplying the circumference by two, we can determine the radius of the base:

12π / 2π = 6

Thus, the base's radius is 6 cm.

Also, we are informed that the cone's volume is 96. As a result, we can get the height using the following formula for a cone's volume:

V = (1/3)r2h

96 = (1/3)(6/2)h

96 = 36 h

96 / 36 = 8/3

Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.

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The probability that the number of homicide deaths for a randomly selected day is:

a) 0: 0.18,

b) 1: 0.35,

c) 2: 0.27

The Poisson distribution is used to simulate the likelihood that a certain number of events will occur within a predetermined window of time or space.

In this instance, the Poisson distribution can be used to simulate the number of homicide deaths that occurred in Richmond, Virginia over the course of a year.

The Poisson distribution can be used to determine the likelihood of 0, 1, and 2 homicides happening on any given day.

One homicide occurs on a randomly chosen day with a 0.18 percent chance, one homicide occurs on a randomly chosen day with a 0.35 percent chance, and two homicides occur on a randomly selected day with a 0.27 percent chance.

Complete Question:

In one year, there were 116 homicide deaths in Richmond, Virginia. Using the Poisson distribution, find the probability that the number of homicide deaths for a randomly selected day is:

a) 0

b) 1

c) 2

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Point G is two, which divides the directed line segment from D to F into a 5:4 ratio at that place.

The general equation for every straight line is y = mx + c, where m is the gradient (or degree of steepness) of the line and c is the y-intercept (the point in which the line crosses the y-axis).

The variables x and y are related to coordinates on the line in the linear equation y = mx + c.

The formula y = mx + c yields a result for y when we enter a value for x.

As y depends on the value of x, it follows that x is an independent variable and y is a dependent variable.

According to our question-

From negative five to positive ten is a number line.

Points D and F are at -2 and +7, respectively.

The distance between point D and F is,

=9

If Point G divides the directed line segment from D to F into a 5: 4 ratio, then,

=2

Hence, The directed line segment from D to F is divided into a 5:4 ratio at point G by the number 2.

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Yes, it is possible for the triangles to be similar. Similar triangles are those whose corresponding sides are proportional to one another. Two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional.

The similarity of two triangles is established if their angles are equal and their sides are proportional in length.

Two triangles are similar if they have the same shape, but not the same size, as their corresponding angles are equal

and the corresponding sides are proportional.

Thus, in the given problem, if the corresponding sides are proportional to each other, then the triangles will be similar.

the side lengths of triangle ABC are 38 , 4x , and 49. The side lengths of triangle DEF are 19, 14 , and 3.5x.

Therefore, by comparing the sides we can find the answer

Let's compare the corresponding sides of the triangles:

AB/DE = 38/19 = 2AC/DF = 49/3.5x = 14BD/EF = 4x/3.5x = 4/3.5 = 8/7

The sides are proportional, so the triangles are similar.

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The final answer is 8arcsin(x/4) - 4sin(2arcsin(x/4)) + C, where C represents the constant of integration. The expression is given in terms of x only.

To evaluate the given integral, we can use trigonometric substitution. Let's substitute x = 4sinθ, which allows us to rewrite the integrand in terms of θ. The differential becomes dx = 4cosθ dθ.

Using this substitution, the integral transforms into ∫(4sinθ)²√(16-(4sinθ)²)(4cosθ) dθ. Simplifying this expression yields 16∫sin²θ√(1-cos²θ)cosθ dθ.

We can apply the double-angle identity sin²θ = (1-cos2θ)/2 to simplify further. This results in 8∫(1-cos2θ)√(1-cos²θ)cosθ dθ.

Next, we can apply the trigonometric identity sin(2θ) = 2sinθcosθ to obtain 8∫(sinθ-sin³θ) dθ.

Finally, integrating term by term and substituting back x = 4sinθ, we arrive at the final answer of 8arcsin(x/4) - 4sin(2arcsin(x/4)) + C. This expression represents the antiderivative of the given function in terms of x only, where C represents the constant of integration.

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The equation of the ellipse will be x² / 36 – y² / 32 = 1.

An ellipse is a locus of a point that moves in a plane such that the sum of its distances from the two points called foci adds up to a constant. It is taken from the cone by cutting it at an angle.

This ellipse is centered at the origin.

Vertices: (0,-6) (0,6)

Foci: (0,-2) (0,2)

Then the equation of the ellipse will be  given as

x² / a² – y² / b² = 1

Let c be the distance between the origin to foci.

Then we have

c = 2

And the value of a will be

a = 6

Then the value of a² will be

a² = 6² = 36

The value of b is the distance from center to one of the Co-vertices. Then the value of b² will be

a² – b² = c²

36 – b² = 2²

        b² = 36 – 4

        b² = 32

Then the equation of the ellipse will be x² / 36 – y² / 32 = 1.

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Answer: The answer is given below

Step-by-step explanation:

Marie's former weight = 175

New weight = 150

Difference = 175 - 150 = 25

Percentage change = 25/175 × 100

= 1/7 × 100

= 14.2%

Jorge's former weight = 190

Jorge's new weight = 180

Difference = 190 - 180 = 10

Percentage change = 10/190 × 100

= 5.26%

From the above solving, the following are right

• Jorge's weight change was approximately 5%.

• Marie's weight change was approximately 14%

• Marie had the greater percent change.

The accurate statements are:

Jorge's weight change was approximately 5%.

Marie's weight change was approximately 14%

. Marie had the greater percent change.

percentage decrease in each person's weight has to be determined:

Percentage decrease =[ (new weight / old weight) -1 ] x 100

The percentage decrease in Marie's weight = (150 / 175) - 1 = 14%

The percentage decrease in Jorge's weight = (180 / 190) - 1 = 5%

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Answer:

D: 27(12x² - 5x - 2).

Step-by-step explanation:

The formula for the surface area of a cylinder is: A = 2πr² + 2πrh

Given that the radius of the cylinder is 3x - 2 cm and the height is x + 3 cm, we can substitute these values in the formula and simplify:

A = 2π(3x - 2)² + 2π(3x - 2)(x + 3)

A = 2π(9x² - 12x + 4) + 2π(3x² + 7x - 6)

A = 18πx² - 24πx + 8π + 6πx² + 14πx - 12π

A = 24πx² - 10πx - 4π

A = 2π(12x² - 5x - 2)

Therefore, the answer is option D: 27(12x² - 5x - 2).

Answer:

D

Step-by-step explanation:

The sum of 8 + 2 ¹ / ₂ would be c. 10 ¹ / ₂.

To calculate the sum of mixed fractions and whole numbers, you can convert the mixed fraction into an improper fraction by adding the numerator to the product of the denominator and the whole number. Then determine a common denominator for both the whole number and the improper fraction.

Finally, simplify the resulting fraction by dividing its numerator and denominator by their greatest common factor.

This means that the sum of 8 and 2 ¹ / ₂ would therefore be :

= 8 + 2

= 10

Then involve the fraction :

= 10 + 1 / 2

= 10 ¹ / ₂

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Using a specific example, we will explore how the cross product behaves under scalar multiplication. In conclusion, the statement does not hold true for all vectors and scalars. The behavior of the cross product under scalar multiplication is not commutative.

To investigate the behavior of the cross product under scalar multiplication, let's consider the vectors a = (1, 2, 3) and b = (4, 5, 6), and a scalar value k = 2.

First, we compute ka × b:

ka × b = 2a × b = 2(1, 2, 3) × (4, 5, 6).

Expanding the cross product, we have:

2(1, 2, 3) × (4, 5, 6) = (2(3×6 - 2×5), 2(1×6 - 3×4), 2(1×5 - 2×4)) = (12, -12, -2).

Next, we compute (ka) × b:

(ka) × b = (2a) × b = (2(1, 2, 3)) × (4, 5, 6).

Expanding the cross product, we have:

(2(1, 2, 3)) × (4, 5, 6) = (2(2×6 - 3×5), 2(3×4 - 1×6), 2(1×5 - 2×4)) = (2, -12, -2).

Finally, we compute a × (kb):

a × (kb) = (1, 2, 3) × (2(4, 5, 6)).

Expanding the cross product, we have:

(1, 2, 3) × (2(4, 5, 6)) = (1×(2×6 - 3×5), 2×(3×4 - 1×6), 3×(1×5 - 2×4)) = (-12, -12, -6).

Comparing the results, we can see that ka × b, (ka) × b, and a × (kb) are not equal in this example. Therefore, it is not true that ka × (b) = (ka) × b = a × (kb) in general.

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Answer:

b. $3,000.00

Step-by-step explanation:

Daniel buys one bond each with a par value of $1,000 from Grath Oil, Ombor Medical Supplies, and Dwyn Horticulture.

Grath Oil bonds are selling at 120.514

Ombor Medical Supplies bonds are selling at 90.773

Dwyn Horticulture bonds are selling at 101.180.

These values have nothing to do here as we only have to find the face value.

The face value of 1 bond is $1000 so, the face value of 3 bonds will be =  dollars.

Answer:

Step-by-step explanation:

a = 0

So that AOB makes a right triangle with sides 3-4-5

Answer:

a = 8

Step-by-step explanation:

Calculate AB using the distance formula and equate to 5

d = \(\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }\)

with (x₁, y₁ ) = A (0, 4 ) and (x₂, y₂ ) = B (3, a )

d = \(\sqrt{(3-0)^2+(a-4)^2}\)

  = \(\sqrt{3^2+(a-4)^2}\)

  = \(\sqrt{9+(a-4)^2}\) , then

\(\sqrt{9+(a-4)^2}\) = 5 ( square both sides )

9 + (a - 4)² = 5² = 25 ( subtract 9 from both sides )

(a - 4)² = 16 ( take square root of both sides )

a - 4 = 4 ( add 4 to both sides )

a = 8

Given the function `f(x, y) = x² - 2xy + 2y`.To find the critical point of the given function, we need to take partial derivatives with respect to x and y and set them equal to zero.

Partial derivative with respect to x `f_x(x,y) = 2x - 2y`Partial derivative with respect to y `f_y(x,y) = -2x + 2`Now, we need to solve these equations for x and y:`f_x(x,y) = 2x - 2y = 0` `=> 2x = 2y` `=> x = y` -------------(1)`f_y(x,y) = -2x + 2 = 0` `=> -2x = -2` `=> x = 1` -------------(2)

Using equations (1) and (2)`y = x = 1`Hence, the critical point of f(x, y) = x² - 2xy + 2y is `(1, 1)`The correct option is a) (1,1).

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Seraphina's speed has increased by a factor of around 23% compared to the first hour.

What Is a Change in Percentage?

The ratio of the difference in the amount to its starting value multiplied by 100 is known as the percentage change. When a number's final value is determined by increasing or decreasing a percentage of its starting value, the percentage change of that quantity will always change.

How can you determine the percentage to the closest tenth?

Rounding to the closest tenth entails adding one integer after the decimal point. The number in the thousandths place, or the second number from the right of the decimal, must be considered while rounding. If the amount is five or more, we add one percent to the number in the tenth position.

Percent Change Formula = \(\frac{ (Final value -Initial value)}{ (Initial value)}\)× 100

Percent Change               = \(\frac{(74-60)}{60}\)× 100

…                                     = \(\frac{14}{60}\) × 100

...                                     ≈ 0.23333 × 100

Percent Change             ≈ 23.33%

The pace of Seraphina increased by around 23% in the second hour compared to the first.

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Answer:

each sticker cost $0.14.

Step-by-step explanation:

To find the cost of each sticker, you can divide the total cost by the number of stickers.

In this case, Minh spent $3.50 on 25 stickers, so the cost of each sticker is:

3.50 / 25 stickers = $0.14/sticker

Therefore, each sticker cost $0.14.

Answer:

Each sticker = $\(0.14\)

Step-by-step explanation:

\(25\) stickers \(=$3.50\)

\(1\) sticker = \(\frac{3.50}{25}\)

\(1\) sticker= \($0.14\)

 The correct option is a. (-8/3, -8/3, -4/3).

To find the farthest point on the sphere x² + y² + z² = 16 from the point (2, 2, 1), we need to find the point on the sphere that is farthest away from the given point. This can be done by considering the distance between the given point and any arbitrary point on the sphere, and then maximizing this distance.

The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) can be calculated using the distance formula:

d = sqrt((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

In this case, we want to maximize the distance between the point (2, 2, 1) and a point on the sphere x² + y² + z² = 16.

Substituting the coordinates of the given point into the distance formula, we have:

d = sqrt((x - 2)² + (y - 2)² + (z - 1)²)

To maximize this distance, we need to maximize the expression inside the square root. Since the sphere equation x² + y² + z² = 16 represents a sphere centered at the origin (0, 0, 0) with radius 4, the farthest point on the sphere from the given point will be diametrically opposite to the given point with respect to the sphere's center.

Therefore, the farthest point on the sphere from the point (2, 2, 1) is the point (-8/3, -8/3, -4/3).

So, the correct option is a. (-8/3, -8/3, -4/3).

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Answer

1

The congruent side must be between the congruent angles to be ASA

"1" has a congruent angle and a congruent side, but there are vertical angles, and these angles are congruent, so it is ASA

Answer:

Solving the equation : \(y+\frac{y}{a}=b\:if\:a\neq\: -1\)  for y we get \(\mathbf{y= \frac{ab}{a+1},\:if\:a\neq -1}\)

Step-by-step explanation:

We need to solve for y the equation: \(y+\frac{y}{a}=b\:if\:a\neq\: -1\)

We need to find value of y

Solving:

\(y+\frac{y}{a}=b\)

Taking LCM of a, 1 we get a

\(\frac{a*y+y}{a}=b\\\frac{ay+y}{a}=b\)

Multiply both sides by a, a will be cancelled on left side

\(a(\frac{ay+y}{a})=ab\\ay+y=ab\)

Taking y common from left side:

\(y(a+1)=ab\)

Divide both sides by a+1

\(\frac{y(a+1)}{a+1}=\frac{ab}{a+1}\\y= \frac{ab}{a+1},\:if\:a\neq -1\)

So, Solving the equation : \(y+\frac{y}{a}=b\:if\:a\neq\: -1\)  for y we get \(\mathbf{y= \frac{ab}{a+1},\:if\:a\neq -1}\)

Answer:

t would be steeper

Step-by-step explanation:

To find the equation of a line parallel or perpendicular to a given line through a point, determine the slope and substitute the point's coordinates into the slope-intercept form.



To find the equation of a line parallel or perpendicular to a given line through a specific point, follow these steps:

1. Determine the slope of the given line. If the given line is in the form y = mx + b, the slope (m) will be the coefficient of x.

2. Parallel Line: A parallel line will have the same slope as the given line. Using the slope-intercept form (y = mx + b), substitute the slope and the coordinates of the given point into the equation to find the new y-intercept (b). This will give you the equation of the parallel line.

3. Perpendicular Line: A perpendicular line will have a slope that is the negative reciprocal of the given line's slope. Calculate the negative reciprocal of the given slope, and again use the slope-intercept form to substitute the new slope and the coordinates of the given point. Solve for the new y-intercept (b) to obtain the equation of the perpendicular line.

Remember that the final equations will be in the form y = mx + b, where m is the slope and b is the y-intercept.Therefore, To find the equation of a line parallel or perpendicular to a given line through a point, determine the slope and substitute the point's coordinates into the slope-intercept form.

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