All practice questions

Question 201Question →

In the coordinate plane, the line representing the linear equation $3x - 4y = 8$ contains the point $(-4, b)$ . What is the value of $b$ ?

Show answer & explanation

Answer: -5

Answer

The value of

b

is

-5

.

Substituting the coordinates

(-4, b)

into the equation

3x - 4y = 8

yields

3(-4) - 4b = 8

. Simplifying gives

-12 - 4b = 8

. Adding

12

to both sides results in

-4b = 20

. Finally, dividing by

-4

gives

b = -5

.

Step-by-Step Solution

1

Substitute the point

(-4, b)

into the equation

3x - 4y = 8

.

3(-4) - 4(b) = 8

Since the point lies on the line, its coordinates must satisfy the line's equation.

2

Simplify the constant term.

-12 - 4b = 8

Multiplying

3

by

-4

yields

-12

.

3

Isolate the variable term by adding

12

to both sides.

-4b = 20

Adding

12

to

8

gives

20

.

4

Solve for

b

by dividing both sides by

-4

.

b = -5

Dividing

20

by

-4

yields

-5

.

Key Concept

Determining an unknown coordinate of a point on a line by substitution into the linear equation.

Question 202Question →

In the standard $(x, y)$ coordinate plane, a circle passes through the points $A(-1, -2)$ and $B(3, 6)$ . The center of the circle, $C$ , lies on the line with the equation $y = 2x - 5$ . What is the radius of this circle?

Show answer & explanation

Answer: 5

Answer

The radius of the circle is 5.

The perpendicular bisector of the segment connecting

A(-1, -2)

and

B(3, 6)

passes through their midpoint

(1, 2)

with a slope of

-\frac{1}{2}

, giving the equation

x + 2y = 5

. Solving the system of equations with

y = 2x - 5

yields the center at

C(3, 1)

. The distance from

C(3, 1)

to

A(-1, -2)

is

\sqrt{(3 - (-1))^2 + (1 - (-2))^2} = \sqrt{4^2 + 3^2} = 5

.

Step-by-Step Solution

1

Find the midpoint and slope of the segment

AB

connecting

A(-1, -2)

and

B(3, 6)

.

Midpoint

M = (1, 2)

and slope

m = 2

.

The center of any circle passing through

A

and

B

must lie on the perpendicular bisector of segment

AB

.

2

Determine the equation of the perpendicular bisector of

AB

.

x + 2y = 5

The perpendicular bisector passes through the midpoint

M(1, 2)

and has a slope that is the negative reciprocal of the slope of

AB

, which is

-\frac{1}{2}

.

3

Find the intersection point of the perpendicular bisector

x + 2y = 5

and the given line

y = 2x - 5

.

Center

C(3, 1)

The center of the circle lies on both the perpendicular bisector of

AB

and the line

y = 2x - 5

.

4

Calculate the distance from the center

C(3, 1)

to point

A(-1, -2)

using the distance formula.

Radius

r = 5

The radius is the distance from the center of the circle to any point on its circumference.

Key Concept

The perpendicular bisector of a chord of a circle passes through the center of that circle.

Question 203Question →

A landscaping company purchases bags of grass seed for $30$ each and bags of fertilizer for $18$ each. For a large project, the company purchased $2$ fewer than $1.5$ times as many bags of fertilizer as bags of grass seed. If the total cost of the grass seed and fertilizer was $420$ , how many bags of fertilizer did the company purchase?

Show answer & explanation

Answer: 10

Answer

The company purchased 10 bags of fertilizer.

The correct answer is 10. By writing the number of fertilizer bags as

f = 1.5g - 2

and substituting this expression into the total cost equation

30g + 18f = 420

, we get

30g + 18(1.5g - 2) = 420

. Simplifying the expression leads to

57g - 36 = 420

, which yields

g = 8

. Substituting

g = 8

back into the relationship for

f

gives

f = 1.5(8) - 2 = 10

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

g

represent the number of grass seed bags and

f

represent the number of fertilizer bags.

Setting up variables is the first step in translating the word problem into solvable algebraic equations.

2

Translate the relationship between the quantities of bags into an equation.

f = 1.5g - 2

The problem states that the number of fertilizer bags purchased is 2 fewer than 1.5 times the number of grass seed bags purchased.

3

Write the total cost equation using the prices and variables.

30g + 18f = 420

The total cost of

420

is the sum of the cost of the grass seed (

30

per bag) and the fertilizer (

18

per bag).

4

Substitute the equation from Step 2 into the cost equation from Step 3.

30g + 18(1.5g - 2) = 420

Substituting allows us to solve a single linear equation with one variable.

5

Distribute the 18 through the parentheses and combine like terms.

57g - 36 = 420

Distributing gives

18 \times 1.5g = 27g

and

18 \times -2 = -36

. Combining

30g

and

27g

yields

57g

.

6

Isolate the variable term by adding 36 to both sides of the equation.

57g = 456

To solve for

g

, we must isolate the term containing the variable.

7

Divide both sides by 57 to find the value of

g

.

g = 8

This division yields the number of grass seed bags purchased.

8

Substitute the value of

g

back into the relation for

f

to find the final answer.

f = 1.5(8) - 2 = 10

The question asks for the number of fertilizer bags (

f

), not grass seed bags (

g

).

Key Concept

Solving a linear equation by substitution and distributing across linear terms.

Estimated Time:1m 30s

Question 204Question →

For all real numbers $x$ such that $x \neq 1$ , the function $f$ is defined by $f(x) = \frac{12}{x - 1}$ . For all real numbers $x$ , the function $g$ is defined by $g(x) = x^2 + 2$ . What is the value of the composite function $f(g(3))$ ?

Show answer & explanation

Answer: 1.2

Answer

1.2

To evaluate the composite function

f(g(3))

, we work from the inside out. First, evaluate the inner function

g(3) = 3^2 + 2 = 11

. Next, substitute this result into the outer function

f(x)

to get

f(11) = \frac{12}{11 - 1} = \frac{12}{10} = 1.2

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = 3

.

g(3) = 11

To evaluate a composite function of the form

f(g(x))

at a given value, we must first find the output of the inner function,

g(x)

, at that value.

2

Substitute the output from Step 1 as the input for the outer function

f(x)

and evaluate.

f(g(3)) = 1.2

Using the result

g(3) = 11

as the input for

f(x)

gives

f(11) = \frac{12}{11 - 1} = 1.2

.

Key Concept

Function Composition and Evaluation

Question 205Question →

For a certain quadratic equation $0.25x^2 - kx + 4.5 = 0$ , where $k$ is a constant, the difference between the two real solutions is exactly $3$ . What is the positive value of $k$ ?

Show answer & explanation

Answer: 2.25

Answer

The positive value of

k

is

2.25

.

By applying the root difference formula

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

with

a = 0.25

,

b = -k

, and

c = 4.5

, we get the equation

\frac{\sqrt{k^2 - 4.5}}{0.25} = 3

. This simplifies to

4\sqrt{k^2 - 4.5} = 3

. Dividing by

4

and squaring both sides gives

k^2 - 4.5 = 0.5625

, which means

k^2 = 5.0625

. Taking the positive square root yields

k = 2.25

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation and express the formula for the difference of the roots.

The coefficients are

a = 0.25

,

b = -k

, and

c = 4.5

. The difference between the roots is

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

.

This sets up the algebraic relationship between the root difference and the coefficients of the quadratic equation.

2

Substitute the coefficients and the given root difference of

3

into the difference formula.

\frac{\sqrt{(-k)^2 - 4(0.25)(4.5)}}{0.25} = 3

This creates an equation in terms of the variable

k

using the given problem constraints.

3

Simplify the equation and isolate the radical term.

4\sqrt{k^2 - 4.5} = 3 \implies \sqrt{k^2 - 4.5} = 0.75

Simplifying the fraction makes it easier to isolate the radical prior to squaring.

4

Square both sides of the equation to eliminate the radical and solve for

k^2

.

k^2 - 4.5 = 0.5625 \implies k^2 = 5.0625

Squaring is the inverse operation of the square root, allowing us to solve for

k^2

.

5

Calculate the positive square root of

5.0625

to find the value of

k

.

k = 2.25

Since the question asks for the positive value of

k

, we choose the positive square root.

Key Concept

Relating the difference of the roots of a quadratic equation to its coefficients using the discriminant and the quadratic formula.

Question 206Question →

A boutique chocolatier packages two types of gift boxes: Standard and Deluxe. Each Standard box contains $4$ dark chocolates, and each Deluxe box contains $8$ dark chocolates. The chocolatier plans to prepare a batch of boxes such that the number of Standard boxes is exactly twice the number of Deluxe boxes. If the chocolatier has a total of $160$ dark chocolates available and uses all of them for this batch, what is the total number of boxes (both Standard and Deluxe combined) they can make?

Show answer & explanation

Answer: 30

Answer

The total number of boxes they can make is 30.

By defining the number of Deluxe boxes as

x

, the number of Standard boxes is

2x

. The equation for the total number of dark chocolates is

4(2x) + 8x = 160

. Solving this gives

16x = 160

, which simplifies to

x = 10

. Thus, there are

10

Deluxe boxes and

20

Standard boxes, for a combined total of

30

boxes.

Step-by-Step Solution

1

Define variables for the number of boxes of each type.

Let

x

be the number of Deluxe boxes and

2x

be the number of Standard boxes.

We are given that the number of Standard boxes is exactly twice the number of Deluxe boxes.

2

Set up an equation representing the total number of dark chocolates used.

4(2x) + 8x = 160

Each Standard box requires

4

dark chocolates, each Deluxe box requires

8

, and the total used is

160

.

3

Solve the equation for

x

.

16x = 160 \Rightarrow x = 10

Combining like terms gives

16x = 160

, and dividing both sides by

16

yields

x = 10

.

4

Calculate the total number of boxes.

x + 2x = 30

The total number of boxes is the sum of Deluxe boxes (

10

) and Standard boxes (

20

).

Key Concept

Translating verbal descriptions of relationships and totals into linear equations to solve algebraic word problems.

Estimated Time:1m 30s

Question 207Question →

A line is defined by the equation $y = 3x + k$ , where $k$ is a constant. This line intersects the parabola $y = x^2 - x + 2$ at two distinct points, $P$ and $Q$ . If the midpoint of the line segment $PQ$ lies on the line $y = 2x + 7$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is

5

.

Equating the equations of the line and the parabola gives a quadratic equation

x^2 - 4x + (2 - k) = 0

. The average of the roots of this quadratic equation gives the

x

-coordinate of the midpoint,

x_m = 2

. Substituting this into the first line's equation gives the

y

-coordinate of the midpoint,

y_m = 6 + k

. Since the midpoint

(2, 6+k)

lies on the line

y = 2x + 7

, we substitute these coordinates to get

6 + k = 11

, which simplifies to

k = 5

.

Step-by-Step Solution

1

Equate the equations of the line and the parabola.

x^2 - 4x + (2 - k) = 0

To find the

x

-coordinates of the intersection points

P

and

Q

.

2

Determine the sum of the

x

-coordinates and find the midpoint's

x

-coordinate.

x_m = 2

By Vieta's formulas, the sum of the roots of the quadratic equation is

4

. The

x

-coordinate of the midpoint is the average of these roots:

4 / 2 = 2

.

3

Find the

y

-coordinate of the midpoint.

y_m = 6 + k

Because the midpoint lies on the line segment

PQ

, its coordinates must satisfy the equation of the line passing through

P

and

Q

, which is

y = 3x + k

.

4

Substitute the midpoint coordinates into the second line's equation and solve for

k

.

k = 5

We are given that the midpoint lies on the line

y = 2x + 7

.

Key Concept

Systems of Linear and Quadratic Equations and Midpoint Properties

Question 208Question →

In the standard $(x,y)$ coordinate plane, a line has an $x$ -intercept of $(a, 0)$ , where $a \neq 0$ , and a $y$ -intercept of $(0, 2a)$ . If the line passes through the point $(4, -3)$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2.5

Answer

The value of

a

is

2.5

.

Representing the intercepts as

(a, 0)

and

(0, 2a)

lets us find the slope of the line, which is

m = \frac{2a - 0}{0 - a} = -2

. The line can then be written as

y = -2x + 2a

. Substituting the point

(4, -3)

into the equation yields

-3 = -2(4) + 2a

, which simplifies to

2a = 5

, and therefore

a = 2.5

.

Step-by-Step Solution

1

Find the slope of the line in terms of the variable

a

.

The slope is

m = -2

.

Applying the slope formula to the points

(a, 0)

and

(0, 2a)

yields

m = \frac{2a - 0}{0 - a} = -2

.

2

Write the general equation of the line.

The equation is

y = -2x + 2a

.

The slope is

-2

and the

y

-intercept is

2a

, so the equation in slope-intercept form is

y = mx + b

.

3

Substitute the coordinates of the point

(4, -3)

to find

a

.

a = 2.5

.

Substituting

x = 4

and

y = -3

into

y = -2x + 2a

gives

-3 = -8 + 2a

, which simplifies to

2a = 5

and

a = 2.5

.

Key Concept

Linear Equations and Graphing

Question 209Question →

In the standard $(x,y)$ coordinate plane, the midpoint of a line segment is $(2, 5)$ . If one of the endpoints of the segment is $(-2, 1)$ , what is the $x$ -coordinate of the other endpoint?

Show answer & explanation

Answer: 6

Answer

The

x

-coordinate of the other endpoint is 6.

According to the midpoint formula, the

x

-coordinate of the midpoint is the average of the

x

-coordinates of the two endpoints. Substituting the given values yields the equation

2 = \frac{-2 + x_2}{2}

. Multiplying both sides by 2 gives

4 = -2 + x_2

, and adding 2 to both sides results in

x_2 = 6

.

Step-by-Step Solution

1

Set up the midpoint equation for the

x

-coordinate using the midpoint formula

x_m = \frac{x_1 + x_2}{2}

.

2 = \frac{-2 + x_2}{2}

The

x

-coordinate of the midpoint is the average of the

x

-coordinates of the endpoints.

2

Multiply both sides of the equation by 2 to solve for the numerator.

4 = -2 + x_2

Multiplying by 2 eliminates the denominator on the right side.

3

Add 2 to both sides of the equation to isolate

x_2

.

x_2 = 6

Adding 2 to both sides isolates the variable

x_2

.

Key Concept

Midpoint Formula

Question 210Question →

In the standard $(x, y)$ coordinate plane, a line $L_1$ with a positive slope $m$ and a negative $y$ -intercept $b$ passes through the point $(4, 3)$ . The region in the fourth quadrant bounded by the line $L_1$ , the $x$ -axis, and the $y$ -axis has an area of exactly $8$ square units. What is the $y$ -coordinate of the $y$ -intercept of line $L_1$ ?

Show answer & explanation

Answer: -6

Answer

The y-coordinate of the y-intercept of line

L_1

is

-6

.

The correct answer is

-6

. Substituting

(4, 3)

into the slope-intercept equation

y = mx + b

gives

3 = 4m + b

, or

m = \frac{3-b}{4}

. The area of the right triangle in the fourth quadrant is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} (-\frac{b}{m})(-b) = \frac{b^2}{2m} = 8

. Substituting

m

yields

b^2 = 16\left(\frac{3-b}{4}\right)

, which simplifies to the quadratic equation

b^2 + 4b - 12 = 0

. Factoring gives

(b+6)(b-2) = 0

. Since the y-intercept

b

must be negative, we have

b = -6

.

Step-by-Step Solution

1

Substitute the given point into the slope-intercept equation

m = \frac{3 - b}{4}

Since the line passes through

(4, 3)

, substituting these coordinates into

y = mx + b

allows us to express the slope

m

in terms of the y-intercept

b

.

2

Determine the intercepts and the dimensions of the bounded region

Base

= -\frac{b}{m}

and Height

= -b

The boundary of the region in the fourth quadrant is a right triangle formed by the origin, the x-intercept

(-\frac{b}{m}, 0)

, and the y-intercept

(0, b)

.

3

Set up the area of the triangle and equate it to 8

b^2 = 16m

The area of a right triangle is

\frac{1}{2} \times \text{base} \times \text{height}

, so

\frac{1}{2} \left(-\frac{b}{m}\right)(-b) = 8

simplifies to

b^2 = 16m

.

4

Substitute

m

into the area equation and solve the resulting quadratic equation

b = -6

(discarding

b = 2

)

Substituting

m = \frac{3 - b}{4}

yields

b^2 + 4b - 12 = 0

, which factors into

(b + 6)(b - 2) = 0

. Since the region is in the fourth quadrant, the y-intercept must be negative (

b < 0

).

Key Concept

Using linear equation intercepts to calculate bounded areas on the coordinate plane and relating variables using point substitution.

Question 211Question →

A parabola in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 - 6x - 8y + 25 = 0$ . What is the distance, in coordinate units, between the focus and the directrix of this parabola?

Show answer & explanation

Answer: 4

Answer

The distance between the focus and the directrix of the parabola is 4.

By completing the square on the equation

x^2 - 6x - 8y + 25 = 0

, we get

(x-3)^2 = 8(y-2)

. Since the coefficient of the linear factor is

8

, we set

4p = 8

, which yields

p = 2

. The distance from the focus to the directrix is

2p = 2(2) = 4

.

Step-by-Step Solution

1

Isolate the terms containing

x

on one side of the equation.

x^2 - 6x = 8y - 25

To set up the equation for completing the square on the

x

terms.

2

Complete the square for the quadratic expression in

x

by adding

9

to both sides.

x^2 - 6x + 9 = 8y - 16 \implies (x-3)^2 = 8y - 16

Adding

( -6/2 )^2 = 9

creates a perfect square trinomial on the left side.

3

Factor out the coefficient of

y

on the right side to write the equation in standard form.

(x-3)^2 = 8(y-2)

This matches the standard form equation

(x-h)^2 = 4p(y-k)

for a vertical parabola.

4

Determine the value of the focal parameter

p

from the standard form.

4p = 8 \implies p = 2

Comparing the standard form coefficient

4p

with the value

8

gives

p = 2

.

5

Calculate the total distance between the focus and the directrix.

2p = 2(2) = 4

The vertex is situated halfway between the focus and the directrix, making the distance between them

2p

.

Key Concept

Finding the geometric properties of a parabola by completing the square to convert its general equation to standard form.

Question 212Question →

The equation of a parabola is given by $(x - 4)^2 = 12(y + 1)$ . What is the $y$ -coordinate of the focus of this parabola?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

The equation

(x - 4)^2 = 12(y + 1)

is a parabola with a vertical axis of symmetry, vertex at

(4, -1)

, and focal length

p = 3

. The focus is located

p

units above the vertex, yielding a

y

-coordinate of

-1 + 3 = 2

.

Step-by-Step Solution

1

Identify the standard form of the parabola's equation.

The equation

(x - 4)^2 = 12(y + 1)

matches

(x - h)^2 = 4p(y - k)

.

This form allows us to find the vertex and the focal distance

p

directly.

2

Determine the vertex and focal distance

p

.

The vertex is

(4, -1)

and

p = 3

since

4p = 12

.

Matching the given equation terms to the standard form reveals these properties.

3

Find the coordinates of the focus.

The focus is at

(4, 2)

.

The focus is located

p

units vertically above the vertex for a parabola opening upward.

Key Concept

Focus of a Parabola

Question 213Question →

What is the product of all real values of $x$ that satisfy the equation $\log_3(x) - 6\log_x(3) = 1$ ?

Show answer & explanation

Answer: 3

Answer

The product of all real values of

x

that satisfy the equation is 3.

By applying the change-of-base formula, the equation becomes

\log_3(x) - \frac{6}{\log_3(x)} = 1

. Substituting

y = \log_3(x)

leads to

y^2 - y - 6 = 0

, which has roots

y = 3

and

y = -2

. These roots correspond to

x = 3^3 = 27

and

x = 3^{-2} = \frac{1}{9}

. Both solutions are valid because they are positive and do not equal 1. The product of these solutions is

27 \times \frac{1}{9} = 3

. Alternatively, using Vieta's formulas, the sum of the roots of the quadratic equation is

y_1 + y_2 = 1

. The product of the solutions is

x_1 x_2 = 3^{y_1} \cdot 3^{y_2} = 3^{y_1 + y_2} = 3^1 = 3

.

Step-by-Step Solution

1

Apply the change-of-base formula to rewrite the variable base term.

\log_3(x) - \frac{6}{\log_3(x)} = 1

This expresses the equation in terms of logarithms with the same base.

2

Use substitution to convert the equation into a quadratic form.

y - \frac{6}{y} = 1 \Rightarrow y^2 - y - 6 = 0

where

y = \log_3(x)

Substitution simplifies the logarithmic equation into a polynomial equation.

3

Solve the quadratic equation by factoring.

(y-3)(y+2) = 0 \Rightarrow y = 3

or

y = -2

Finding the roots for

y

is the intermediate step to solving for

x

.

4

Back-substitute to find the values of

x

.

x = 3^3 = 27

and

x = 3^{-2} = \frac{1}{9}

Converting from logarithmic form back to exponential form yields the values of

x

.

5

Multiply the solutions together.

27 \times \frac{1}{9} = 3

The question asks for the product of all real solutions.

Key Concept

Solving logarithmic equations using the change-of-base formula and quadratic substitution.

Alternative Method

Instead of solving for individual values of

x

, note that if

y_1

and

y_2

are the roots of the quadratic equation

y^2 - y - 6 = 0

, then

y_1 + y_2 = 1

by Vieta's formulas. Since

x_1 = 3^{y_1}

and

x_2 = 3^{y_2}

, the product of the solutions is

x_1 x_2 = 3^{y_1} \cdot 3^{y_2} = 3^{y_1 + y_2} = 3^1 = 3

.

Estimated Time:2m 0s

Question 214Question →

In $\triangle ABC$ , the measure of exterior angle $\angle BCD$ is $115^\circ$ . If the measure of interior angle $\angle A$ is $45^\circ$ , what is the measure, in degrees, of interior angle $\angle B$ ?

Show answer & explanation

Answer: 70

Answer

The measure of interior angle

\angle B

is

70

degrees.

By the Exterior Angle Theorem, the measure of exterior angle

\angle BCD

is equal to the sum of the two remote interior angles,

\angle A

and

\angle B

. We can write this relationship as

m\angle BCD = m\angle A + m\angle B

. Substituting

115^\circ

for

m\angle BCD

and

45^\circ

for

m\angle A

gives

115 = 45 + m\angle B

. Solving for

m\angle B

yields

70^\circ

.

Step-by-Step Solution

1

Set up the equation using the Exterior Angle Theorem.

m\angle BCD = m\angle A + m\angle B

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

2

Substitute the given measurements into the equation.

115 = 45 + m\angle B

The exterior angle

\angle BCD

measures

115^\circ

and the remote interior angle

\angle A

measures

45^\circ

.

3

Solve for the unknown angle measure by subtraction.

m\angle B = 70

Subtracting

45

from both sides isolates

m\angle B

.

Key Concept

Exterior Angle Theorem

Question 215Question →

In the standard $(x, y)$ coordinate plane, a line passes through the point $(3, 2)$ and has a $y$ -intercept of $-4$ . If the point $(k, 5k + 2)$ also lies on this line, what is the value of $k$ ?

Show answer & explanation

Answer: -2

Answer

The value of

k

is

-2

.

The line has a slope of

2

and a y-intercept of

-4

, giving the equation

y = 2x - 4

. Substituting the coordinates of

(k, 5k + 2)

results in

5k + 2 = 2k - 4

, which simplifies to

3k = -6

, yielding

k = -2

.

Step-by-Step Solution

1

Identify the coordinates of the y-intercept.

The y-intercept of

-4

corresponds to the point

(0, -4)

.

The y-intercept is the point where the line crosses the y-axis, meaning the x-coordinate is 0.

2

Calculate the slope of the line.

The slope

m

is

2

.

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the points

(3, 2)

and

(0, -4)

gives

m = \frac{2 - (-4)}{3 - 0} = \frac{6}{3} = 2

.

3

Write the equation of the line.

The equation of the line is

y = 2x - 4

.

Using the slope-intercept form

y = mx + b

, where the slope

m = 2

and the y-intercept

b = -4

.

4

Substitute the point

(k, 5k + 2)

into the line's equation.

The equation becomes

5k + 2 = 2k - 4

.

Since the point lies on the line, its coordinates must satisfy the line's equation.

5

Solve the linear equation for

k

.

k = -2

.

Subtracting

2k

from both sides gives

3k + 2 = -4

. Subtracting

2

from both sides gives

3k = -6

. Dividing by

3

gives

k = -2

.

Key Concept

Finding the equation of a line from a point and an intercept, and solving for parameters of points on that line.

Question 216Question →

Two cyclists start at opposite ends of a $90$ -mile trail at the same time and ride toward each other. One cyclist rides at a constant speed that is $3$ miles per hour faster than the other cyclist. If the two cyclists meet after exactly $3$ hours, what is the constant speed, in miles per hour, of the faster cyclist?

Show answer & explanation

Answer: 16.5

Answer

The speed of the faster cyclist is

16.5

miles per hour.

The correct answer of

16.5

is found by setting the speed of the slower cyclist to

s

and the faster cyclist to

s + 3

. Since both cyclists ride toward each other for

3

hours, their combined distance is

3s + 3(s + 3) = 90

. Solving for

s

yields

6s + 9 = 90

, which simplifies to

6s = 81

, or

s = 13.5

. Adding

3

to

13.5

gives the speed of the faster cyclist, which is

16.5

miles per hour.

Step-by-Step Solution

1

Define variables for the speeds of both cyclists in terms of a single variable.

Let

s

be the speed of the slower cyclist. The speed of the faster cyclist is

s + 3

.

Using a single variable simplifies the setup of a solvable linear equation.

2

Write a linear equation using the relationship

\text{Distance} = \text{Speed} \times \text{Time}

.

The equation is

3s + 3(s + 3) = 90

.

The sum of the distances traveled by both cyclists when they meet must equal the total length of the trail, which is

90

miles.

3

Solve the equation for

s

.

6s + 9 = 90 \implies 6s = 81 \implies s = 13.5

.

This yields the speed of the slower cyclist.

4

Calculate the speed of the faster cyclist.

13.5 + 3 = 16.5

.

The question specifically asks for the speed of the faster cyclist, which is represented by

s + 3

.

Key Concept

Translating distance-rate-time relationships from word problems into solvable linear equations.

Alternative Method

An alternative approach is to use the concept of relative speed. Since the two cyclists are moving directly toward each other, their relative speed of approach is the sum of their individual speeds. They cover a total of

90

miles in

3

hours, which means their combined speed is

\frac{90}{3} = 30

miles per hour. If the speed of the faster cyclist is

f

and the slower is

s

, then

f + s = 30

and

f - s = 3

. Adding these two equations gives

2f = 33

, which yields

f = 16.5

miles per hour.

Estimated Time:1m 15s

Question 217Question →

In the standard $(x, y)$ coordinate plane, an ellipse is defined by the equation $7x^2 + 16y^2 - 42x - 32y - 33 = 0$ . A parabola has its vertex at the focus of the ellipse with the smaller $x$ -coordinate, and its focus at the focus of the ellipse with the larger $x$ -coordinate. What is the larger of the two $y$ -coordinates of the points on the parabola that have an $x$ -coordinate of 6?

Show answer & explanation

Answer: 13

Answer

The larger of the two

y

-coordinates of the points on the parabola is 13.

By completing the square on the general ellipse equation, we get

\frac{(x-3)^2}{16} + \frac{(y-1)^2}{7} = 1

. The center is

(3, 1)

and the focal distance is

c = \sqrt{16-7} = 3

, meaning the foci are at

(0, 1)

and

(6, 1)

. The parabola has its vertex at

(0, 1)

and focus at

(6, 1)

, which means it opens to the right with

p = 6

. Its equation is

(y - 1)^2 = 24x

. Substituting

x = 6

yields

(y - 1)^2 = 144

, so

y - 1 = \pm 12

. The two possible

y

-coordinates are

13

and

-11

, of which

13

is the larger value.

Step-by-Step Solution

1

Complete the square for the given ellipse equation to rewrite it in standard form.

\frac{(x-3)^2}{16} + \frac{(y-1)^2}{7} = 1

Converting the equation to standard form is necessary to determine the center and semi-axis lengths of the ellipse.

2

Find the focal distance

c

and calculate the coordinates of the foci.

Focal distance

c = 3

; Foci at

(0, 1)

and

(6, 1)

For an ellipse, the distance

c

from the center

(h, k)

to the foci is

\sqrt{a^2 - b^2}

. Since the major axis is horizontal, the foci are located at

(h \pm c, k)

.

3

Use the foci coordinates to identify the vertex and focus of the parabola.

Vertex:

(0, 1)

; Focus:

(6, 1)

The problem defines the parabola's vertex as the ellipse focus with the smaller

x

-coordinate, and the parabola's focus as the ellipse focus with the larger

x

-coordinate.

4

Determine the equation of the parabola using its vertex and focus.

(y - 1)^2 = 24x

The parabola is horizontal and opens to the right with focal distance

p = 6

. The standard form is

(y - k)^2 = 4p(x - h)

.

5

Substitute

x = 6

into the parabola equation and solve for the larger

y

-value.

y = 13

Substituting

x = 6

yields

(y - 1)^2 = 144

, which gives

y = 1 + 12 = 13

or

y = 1 - 12 = -11

. The larger value is 13.

Key Concept

Determining the equations and key features (foci, vertices, focal parameters) of ellipses and parabolas by rewriting equations into standard forms.

Alternative Method

Once the equation

(y - 1)^2 = 24x

is established, recognize that at

x = 6

(which is the

x

-coordinate of the focus), the points on the parabola form the endpoints of the latus rectum. The length of the latus rectum is

4p = 24

, so the points lie at distance

2p = 12

vertically above and below the focus

(6, 1)

. Thus, the

y

-coordinates are

1 \pm 12

, immediately yielding the larger coordinate as 13.

Estimated Time:3m 0s

Question 218Question →

A line graphed in the standard $(x,y)$ coordinate plane has an $x$ -intercept of $6$ and a $y$ -intercept of $-3$ . What is the slope of this line?

Show answer & explanation

Answer: 0.5

Answer

The slope of the line is

0.5

(or

\frac{1}{2}

).

The correct slope is

0.5

. By identifying the

x

-intercept as

(6, 0)

and the

y

-intercept as

(0, -3)

, we can apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to find

m = \frac{-3 - 0}{0 - 6} = \frac{-3}{-6} = 0.5

.

Step-by-Step Solution

1

Identify the coordinates of the intercepts on the coordinate plane.

The points are

(6, 0)

and

(0, -3)

.

An

x

-intercept of

6

means the line crosses the

x

-axis at

(6, 0)

. A

y

-intercept of

-3

means the line crosses the

y

-axis at

(0, -3)

.

2

Apply the slope formula with the identified coordinates.

m = \frac{-3 - 0}{0 - 6} = \frac{-3}{-6} = 0.5

The slope formula is defined as

m = \frac{y_2 - y_1}{x_2 - x_1}

for any two points on a line.

Key Concept

Calculating the slope of a line from its intercepts

Question 219Question →

In the standard $(x, y)$ coordinate plane, the points $A(1, 2)$ and $B(9, 8)$ are the endpoints of a diameter of a circle $C$ . A line $L$ passes through the center of $C$ and is perpendicular to segment $AB$ . A point $P(x, y)$ lies on line $L$ such that the distance from $P$ to the center of $C$ is equal to the radius of $C$ . If the $x$ -coordinate of $P$ is greater than the $x$ -coordinate of the center of $C$ , what is the $y$ -coordinate of $P$ ?

Show answer & explanation

Answer: 1

Answer

The y-coordinate of the point P is 1.

The correct answer is 1. The center of circle C is the midpoint of the diameter AB, which is calculated as M(5, 5). The radius is half the length of AB, which is 5. The line L passing through M perpendicular to AB has a slope of -4/3. Points on this line at a distance of 5 from M are found by changing the coordinates by (+3, -4) or (-3, +4), yielding (8, 1) and (2, 9). Since the x-coordinate must be greater than the center's x-coordinate of 5, the correct point is (8, 1), which has a y-coordinate of 1.

Step-by-Step Solution

1

Calculate the center of the circle C by finding the midpoint of the diameter AB.

The center is M(5, 5).

The center of a circle is the midpoint of any of its diameters.

2

Calculate the radius of circle C by finding half the distance between A(1, 2) and B(9, 8).

The radius is 5.

The distance formula gives the diameter length as 10, and the radius is half the diameter.

3

Find the slope of line L perpendicular to AB.

The slope of L is -4/3.

The slope of AB is 3/4, and perpendicular lines have slopes that are negative reciprocals of each other.

4

Determine the coordinates of point P using the distance from the center and the slope of line L.

The possible points are (8, 1) and (2, 9).

Moving a distance of 5 along a line with slope -4/3 from (5, 5) results in a change of +/-3 in the x-coordinate and -/+4 in the y-coordinate.

5

Apply the constraint that the x-coordinate of P must be greater than the x-coordinate of the center (5).

P is (8, 1), so the y-coordinate is 1.

Comparing the two candidate points, only (8, 1) has an x-coordinate greater than 5.

Key Concept

Applying midpoint, distance, and perpendicular slope relationships in coordinate geometry to locate points.

Estimated Time:2m 30s

Question 220Question →

For a certain real number $x$ , the equation $0.4(3x - 5) - 0.15(2x + 8) = 1.3$ is true. What is the value of $4x - 3$ ?

Show answer & explanation

Answer: 17

Answer

The value of the expression is 17.

Distributing the decimals results in

1.2x - 2 - 0.3x - 1.2 = 1.3

. Combining like terms yields

0.9x - 3.2 = 1.3

. Adding

3.2

to both sides results in

0.9x = 4.5

, which simplifies to

x = 5

after dividing by

0.9

. Substituting

x = 5

into the expression

4x - 3

gives

4(5) - 3 = 17

.

Step-by-Step Solution

1

Distribute the decimal factors through the parentheses on the left side of the equation.

1.2x - 2 - 0.3x - 1.2 = 1.3

To eliminate the parentheses and set up terms for simplification.

2

Combine the variable terms and constant terms on the left side of the equation.

0.9x - 3.2 = 1.3

To group like terms and simplify the equation.

3

Isolate the variable term by adding

3.2

to both sides of the equation.

0.9x = 4.5

To gather all constant terms on the right side of the equation.

4

Divide both sides of the equation by

0.9

to solve for

x

.

x = 5

To find the numerical value of the variable.

5

Substitute the value of

x

into the requested expression

4x - 3

.

4(5) - 3 = 17

To evaluate the specific expression requested by the question.

Key Concept

Solving multi-step linear equations with decimals and evaluating algebraic expressions

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