All practice questions

If $3(k + 4) = 5k - 8$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

10

Distributing the

3

on the left side yields

3k + 12 = 5k - 8

. Subtracting

3k

from both sides yields

12 = 2k - 8

. Adding

8

to both sides yields

20 = 2k

. Dividing by

2

gives the solution

k = 10

.

Step-by-Step Solution

1

Distribute

3

to the terms inside the parentheses

3k + 12 = 5k - 8

To simplify the left side of the equation

2

Subtract

3k

from both sides of the equation

12 = 2k - 8

To group the variable terms on one side

3

Add

8

to both sides of the equation

20 = 2k

To isolate the variable term

4

Divide both sides by

2

k = 10

To solve for

k

Key Concept

Solving linear equations with variables on both sides using distributive property

Question 42Question →

In the equation $\frac{3}{4}(8x - 12) + kx = 10x - 9$ , $k$ is a constant. If the equation has infinitely many solutions, what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Distributing the fraction on the left side of the equation yields

\frac{3}{4}(8x) - \frac{3}{4}(12) = 6x - 9

. Substituting this back gives

6x - 9 + kx = 10x - 9

. Factoring out

x

on the left side gives

(6 + k)x - 9 = 10x - 9

. For a linear equation to have infinitely many solutions, the coefficients of

x

on both sides must be identical, and the constants must be identical. Since the constants on both sides are already

-9

, we set the coefficients equal:

6 + k = 10

. Subtracting 6 from both sides gives the correct value

k = 4

.

Step-by-Step Solution

1

Distribute the fraction

\frac{3}{4}

to the terms inside the parentheses.

6x - 9 + kx = 10x - 9

To simplify the expression and eliminate the parentheses.

2

Factor out

x

from the terms on the left side of the equation.

(6 + k)x - 9 = 10x - 9

To group the

x

terms together to easily compare coefficients.

3

Set the coefficient of

x

on the left side equal to the coefficient of

x

on the right side.

6 + k = 10

For the equation to have infinitely many solutions, the coefficients of the variable on both sides must be equal when the constant terms are equal.

4

Solve for

k

by subtracting 6 from both sides of the equation.

k = 4

To isolate the constant

k

.

Key Concept

A linear equation in one variable has infinitely many solutions when it can be simplified to an identity of the form

Ax + B = Ax + B

, meaning both the coefficients of

x

and the constant terms on both sides of the equation are equal.

Question 43Question →

In the equation below, $a$ and $b$ are constants.

\frac{1}{3}(2a - 5x) - \frac{3}{4}(x - b) = -\frac{29}{12}x + 5

If the equation has infinitely many solutions for $x$ , what is the value of $8a + 9b$ ?

Show answer & explanation

Answer: 60

Answer

The value of

8a + 9b

is

60

.

To find the value of

8a + 9b

that makes the equation have infinitely many solutions, we first expand and simplify the left side of the equation:

\frac{2}{3}a - \frac{5}{3}x - \frac{3}{4}x + \frac{3}{4}b = -\frac{29}{12}x + 5

. Combining the

x

terms gives

-\frac{29}{12}x + \(\frac{2}{3}a + \frac{3}{4}b\) = -\frac{29}{12}x + 5

. Since the coefficients of

x

on both sides are equal (

-\frac{29}{12}

), the equation will have infinitely many solutions if the constant terms on both sides are also equal. This requires

\frac{2}{3}a + \frac{3}{4}b = 5

. Multiplying this entire equation by the least common multiple of the denominators, which is 12, yields

8a + 9b = 60

.

Step-by-Step Solution

1

Distribute the constants through the parentheses on the left side of the equation.

\frac{2}{3}a - \frac{5}{3}x - \frac{3}{4}x + \frac{3}{4}b = -\frac{29}{12}x + 5

This separates the variable terms from the constant terms so the equation can be simplified.

2

Combine the coefficients of the

x

terms on the left side using 12 as the common denominator.

-\frac{29}{12}x + \frac{2}{3}a + \frac{3}{4}b = -\frac{29}{12}x + 5

Simplifying the variable terms allows us to compare the coefficients on both sides of the equation.

3

Equate the constant terms from the left and right sides of the equation.

\frac{2}{3}a + \frac{3}{4}b = 5

A linear equation has infinitely many solutions when the coefficients of the variable on both sides are equal and the constant terms on both sides are also equal.

4

Multiply both sides of the equation by 12 to eliminate the fractional denominators.

8a + 9b = 60

Multiplying the equation by the common denominator directly evaluates the target expression

8a + 9b

.

Key Concept

For a linear equation in one variable to have infinitely many solutions, it must be reducible to an identity of the form

cx + d = cx + d

, where both the variable coefficients and the constant terms on both sides of the equation are equal.

Question 44Question →

A factory manufactures solar panels at a constant rate. At the start of a morning shift, the factory has already manufactured $80$ solar panels. The total number of solar panels manufactured $t$ hours after the shift begins is modeled by a linear function. If the factory has manufactured a total of $200$ solar panels $3$ hours after the shift begins, how many hours after the shift begins will the factory have manufactured a total of $440$ solar panels?

Show answer & explanation

Answer: 9

Answer

9

To find the time when the total number of solar panels reaches

440

, we first determine the constant rate of production. The change in solar panels over the first

3

hours is

200 - 80 = 120

panels. Dividing by the

3

hours gives a constant rate of

40

panels per hour. The linear equation representing the total panels is

N(t) = 40t + 80

. Setting

N(t) = 440

gives

440 = 40t + 80

, which simplifies to

360 = 40t

. Solving for

t

yields

t = 9

.

Step-by-Step Solution

1

Calculate the constant rate of production (slope) of the linear function.

The rate is

40

solar panels per hour.

The factory starts with

80

solar panels and reaches

200

solar panels in

3

hours. The rate of change is the change in the number of panels divided by the change in time:

\frac{200 - 80}{3 - 0} = \frac{120}{3} = 40

panels per hour.

2

Write the linear equation representing the total number of solar panels manufactured,

N(t)

, after

t

hours.

N(t) = 40t + 80

Since the initial quantity is

80

and the rate of production is

40

panels per hour, the linear function is

N(t) = 40t + 80

.

3

Solve for the time

t

when the total number of solar panels is

440

.

t = 9

hours

Set

N(t) = 440

in the equation:

440 = 40t + 80

. Subtract

80

from both sides to get

360 = 40t

. Divide both sides by

40

to find

t = 9

.

Key Concept

Linear Functions and Graphs

Estimated Time:1m 30s

Question 45Question →

In the equation $\frac{1}{2}(kx - 4) - \frac{2}{3}(x - k) = \frac{5}{6}x - 1$ , where $k$ is a constant, the equation has no solution for $x$ . What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of the constant

k

for which the equation has no solution is

3

.

For a linear equation of the form

Ax + B = Cx + D

to have no solution, the coefficients of the variable terms must be equal (

A = C

) but the constant terms must be unequal (

B \neq D

). Distributing the terms on the left side of the given equation and grouping them yields

\left(\frac{1}{2}k - \frac{2}{3}\right)x + \left(\frac{2}{3}k - 2\right) = \frac{5}{6}x - 1

. Setting the coefficients of

x

equal to each other gives

\frac{1}{2}k - \frac{2}{3} = \frac{5}{6}

. Adding

\frac{2}{3}

to both sides yields

\frac{1}{2}k = \frac{9}{6}

, or

\frac{1}{2}k = \frac{3}{2}

, which simplifies to

k = 3

. Evaluating the constant terms when

k = 3

gives

\frac{2}{3}(3) - 2 = 0

on the left side and

-1

on the right side. Since

0 \neq -1

, the equation has no solution when

k = 3

.

Step-by-Step Solution

1

Distribute the fractional coefficients on the left side of the equation.

\frac{1}{2}kx - 2 - \frac{2}{3}x + \frac{2}{3}k = \frac{5}{6}x - 1

To separate the variable terms from the constants for further algebraic manipulation.

2

Group the terms on the left side into a coefficient for

x

and a single constant term.

\left(\frac{1}{2}k - \frac{2}{3}\right)x + \left(\frac{2}{3}k - 2\right) = \frac{5}{6}x - 1

To represent the equation in the standard linear format

Ax + B = Cx + D

.

3

Set the coefficients of

x

from both sides equal to each other.

\frac{1}{2}k - \frac{2}{3} = \frac{5}{6}

A linear equation has no solution when the variable terms on both sides cancel out, requiring their coefficients to be identical.

4

Solve the linear equation to isolate the constant

k

.

k = 3

Add

\frac{2}{3}

to both sides to get

\frac{1}{2}k = \frac{9}{6}

, which simplifies to

\frac{1}{2}k = \frac{3}{2}

. Multiplying both sides by

2

yields

k = 3

.

5

Verify that the constant terms are not equal when substituting

k = 3

.

The left-side constant is

0

and the right-side constant is

-1

. Since

0 \neq -1

, the condition is satisfied.

To ensure the equation does not simplify to an identity with infinitely many solutions (which occurs when both the variable coefficients and the constant terms are equal).

Key Concept

Determining the conditions under which a linear equation in one variable has no solution.

Alternative Method

To avoid working with fractions, multiply every term in the equation by

6

(the least common multiple of

2, 3, 6

) at the start:

3(kx - 4) - 4(x - k) = 5x - 6

. Expand the parentheses to get

3kx - 12 - 4x + 4k = 5x - 6

, and group the terms:

(3k - 4)x + (4k - 12) = 5x - 6

. For there to be no solution, set the coefficients of

x

equal to each other:

3k - 4 = 5 \implies 3k = 9 \implies k = 3

. Check the constant terms with

k = 3

:

4(3) - 12 = 0

, which is unequal to

-6

. This confirms

k = 3

is the correct answer.

Estimated Time:2m 0s

Question 46Question →

In the $xy$ -plane, the graph of the linear function $f(x) = px + q$ , where $p$ and $q$ are constants, passes through the point $(2, 7)$ . Line $L$ is parallel to the graph of $f$ and has a $y$ -intercept that is $3$ units below the $y$ -intercept of the graph of $f$ . If the $x$ -intercept of line $L$ is $(-6, 0)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

p

is

0.5

(or the fraction

1/2

).

The correct answer is

0.5

(or

1/2

). The graph of

f(x) = px + q

passes through

(2, 7)

, which means

7 = 2p + q

, or

q = 7 - 2p

. Line

L

is parallel to

f

, so its slope is

p

, and its

y

-intercept is

q - 3

. Thus, the equation of line

L

is

y = px + q - 3

. Since line

L

has an

x

-intercept at

(-6, 0)

, we can substitute

x = -6

and

y = 0

into its equation, yielding

0 = -6p + q - 3

. Substituting

q = 7 - 2p

into this equation gives

0 = -6p + (7 - 2p) - 3

, which simplifies to

-8p + 4 = 0

. Solving for

p

gives

8p = 4

, or

p = 0.5

.

Step-by-Step Solution

1

Express the relationship between

p

and

q

using the given point

(2, 7)

on the graph of

f

.

q = 7 - 2p

The point

(2, 7)

must satisfy the equation

f(x) = px + q

.

2

Formulate the equation of line

L

using the parallel slope and the shifted

y

-intercept.

y = px + q - 3

Parallel lines have equal slopes, and the

y

-intercept of

L

is

3

units below the

y

-intercept of

f

, which is

q

.

3

Substitute the

x

-intercept

(-6, 0)

into the equation of line

L

.

-6p + q - 3 = 0

The

x

-intercept is a point on the line where

y = 0

.

4

Substitute the expression for

q

from Step 1 into the equation from Step 3 and solve for

p

.

p = 0.5

(or

\frac{1}{2}

)

Solving the linear equation

-6p + (7 - 2p) - 3 = 0

simplifies to

-8p + 4 = 0

, giving

p = 0.5

.

Key Concept

Understanding linear functions, their graphs, slopes of parallel lines, and intercepts.

Alternative Method

Instead of solving for

q

first, you can use the point-slope form. Line

L

passes through

(-6, 0)

and has slope

p

, so its equation is

y = p(x + 6)

, or

y = px + 6p

. The

y

-intercept of

L

is

6p

. The

y

-intercept of the graph of

f

is

q

. We are given that the

y

-intercept of

L

is

3

units below the

y

-intercept of

f

, so

6p = q - 3

. Since

f

passes through

(2, 7)

, we have

7 = 2p + q

, which means

q = 7 - 2p

. Substituting this into

6p = q - 3

gives

6p = (7 - 2p) - 3

, or

6p = 4 - 2p

. Adding

2p

to both sides gives

8p = 4

, which results in

p = 0.5

.

Estimated Time:2m 30s

Question 47Question →

A rental company charges a flat fee of $35$ dollars plus $12$ dollars per hour to rent a bicycle. If a customer was charged a total of $95$ dollars, for how many hours did the customer rent the bicycle?

Show answer & explanation

Answer: 5

Answer

5

To find the number of hours the bicycle was rented, we can set up the equation

12h + 35 = 95

, where

h

is the number of hours. Subtracting

35

from both sides gives

12h = 60

. Dividing both sides by

12

gives

h = 5

. Thus, the customer rented the bicycle for

5

hours.

Step-by-Step Solution

1

Set up a linear equation representing the total cost.

12h + 35 = 95

, where

h

represents the number of hours rented.

The total cost consists of a flat fee of

35

dollars plus an hourly charge of

12

dollars multiplied by the number of hours

h

.

2

Subtract the flat fee from both sides of the equation to isolate the variable term.

12h = 60

Subtracting

35

from both sides simplifies the equation to find the total hourly charge portion of the cost.

3

Divide by the hourly rate to solve for the number of hours

h

.

h = 5

Dividing both sides by

12

isolates

h

to find the number of rental hours.

Key Concept

Linear Equations in One Variable

Question 48Question →

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $-3$ and passes through the point $(4, 18)$ . What is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 30

Answer

The correct answer is 30.

The equation of a linear function can be written in slope-intercept form as

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-coordinate of the

y

-intercept. Substituting the given slope

m = -3

yields

f(x) = -3x + b

. Since the point

(4, 18)

lies on the graph of

f

, we substitute

x = 4

and

f(x) = 18

into the equation to get

18 = -3(4) + b

, which simplifies to

18 = -12 + b

. Adding

12

to both sides of the equation yields

b = 30

. Thus, the

y

-coordinate of the

y

-intercept of the graph of

f

is

30

.

Step-by-Step Solution

1

Write the general slope-intercept form of a linear equation.

f(x) = -3x + b

The slope of the line is given as

-3

, so we substitute

m = -3

into

f(x) = mx + b

.

2

Substitute the point

(4, 18)

into the equation.

18 = -3(4) + b

Since the graph of

f

passes through

(4, 18)

, these coordinates must satisfy the function equation.

3

Solve for the

y

-intercept

b

.

b = 30

Simplify to

18 = -12 + b

, and then add

12

to both sides to isolate

b

.

Key Concept

Finding the equation of a line using its slope and a point.

Question 49Question →

In the equation below, $m$ is a constant.

\frac{3}{5}(5x - 10) - 2(x - m) = 18

If the solution to the equation is $x = 12$ , what is the value of $m$ ?

Show answer & explanation

Answer: 6

Answer

The correct value of

m

is 6.

Substituting

x = 12

into the equation yields

\frac{3}{5}(5(12) - 10) - 2(12 - m) = 18

, which simplifies to

30 - 24 + 2m = 18

. Combining constant terms gives

6 + 2m = 18

. Subtracting 6 from both sides gives

2m = 12

, and dividing by 2 yields

m = 6

.

Step-by-Step Solution

1

Substitute

x = 12

into the given equation.

\frac{3}{5}(5(12) - 10) - 2(12 - m) = 18

Since

x = 12

is the solution, it must satisfy the equation.

2

Simplify the expression inside the first set of parentheses.

5(12) - 10 = 60 - 10 = 50

Evaluate the terms inside the parentheses first.

3

Multiply the simplified term by the fraction

\frac{3}{5}

.

\frac{3}{5}(50) = 30

Multiply 50 by 3 and divide by 5.

4

Substitute 30 back into the equation and distribute

-2

to the terms inside the second set of parentheses.

30 - 24 + 2m = 18

Distributing

-2

to

12 - m

yields

-24 + 2m

due to sign rules.

5

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

6 + 2m = 18

30 - 24 = 6

.

6

Subtract 6 from both sides to isolate the term with

m

.

2m = 12

Isolate the variable term on one side of the equation.

7

Divide both sides by 2 to solve for

m

.

m = 6

Divide 12 by 2 to find the final value.

Key Concept

Solving a linear equation in one variable by substitution and isolation.

Question 50Question →

In the $xy$ -plane, line $k$ has the equation $y = -2x + 10$ . Line $m$ is parallel to line $k$ and passes through the point $(6, 8)$ . Line $l$ is perpendicular to line $k$ and intersects the $x$ -axis at the point $(a, 0)$ , where $a > 10$ . If the region bounded by lines $k$ , $m$ , $l$ , and the $y$ -axis has an area of $110$ , what is the value of $a$ ?

Show answer & explanation

Answer: 25

Answer

25

To find the value of

a

, we determine the equations of the lines

m

and

l

based on their geometric relationships to line

k

. Line

m

is parallel to line

k

(

y = -2x + 10

), so its slope is

-2

. Using the point

(6, 8)

, its equation is

y = -2x + 20

. Line

l

is perpendicular to line

k

, so its slope is

\frac{1}{2}

. It intersects the

x

-axis at

(a, 0)

, giving the equation

y = \frac{1}{2}x - \frac{1}{2}a

. The bounded region formed by the parallel lines

k

and

m

, the perpendicular line

l

, and the

y

-axis is a trapezoid. Calculating the area of this trapezoid by dividing it into a parallelogram and a triangle yields the area formula

60 + 2a

. Setting this equal to the given area of

110

yields

60 + 2a = 110

, which solves to

a = 25

.

Step-by-Step Solution

1

Determine the equation of line

m

using the parallel slope and the given point.

Line

m

has the equation

y = -2x + 20

.

Parallel lines have equal slopes. Since line

k

has a slope of

-2

, line

m

also has a slope of

-2

. Substituting the point

(6, 8)

into the point-slope form gives

y - 8 = -2(x - 6)

.

2

Determine the equation of line

l

using the perpendicular slope and its

x

-intercept.

Line

l

has the equation

y = \frac{1}{2}x - \frac{1}{2}a

.

Perpendicular lines have negative reciprocal slopes. The negative reciprocal of

-2

is

\frac{1}{2}

. Using the point

(a, 0)

in the point-slope form gives

y - 0 = \frac{1}{2}(x - a)

.

3

Calculate the vertices of the bounded region by finding the intersection points of the boundary lines.

The vertices of the bounded region are

(0, 20)

,

(0, 10)

,

(4 + 0.2a, 2 - 0.4a)

, and

(8 + 0.2a, 4 - 0.4a)

.

The region is bounded by the parallel lines

k

and

m

, the perpendicular line

l

, and the

y

-axis (

x = 0

).

4

Find the area of the region as an algebraic expression in terms of

a

.

The area is equal to

60 + 2a

.

The region can be divided into a parallelogram with a vertical base of

10

and width

4 + 0.2a

, and a right triangle with a vertical base of

10

and width

4

. The sum of their areas is

10(4 + 0.2a) + \frac{1}{2}(10)(4) = 40 + 2a + 20 = 60 + 2a

.

5

Set the area expression equal to the given area of

110

and solve for

a

.

a = 25

Setting the area equal to

110

yields

60 + 2a = 110

, which simplifies to

2a = 50

, or

a = 25

.

Key Concept

Linear functions, parallel and perpendicular lines, finding line equations, and coordinate geometry area.

Alternative Method

The area can also be calculated using the geometric properties of a trapezoid. The height of the trapezoid is the perpendicular distance between the parallel lines

k

and

m

, which is

\frac{|20 - 10|}{\sqrt{(-2)^2 + 1^2}} = 2\sqrt{5}

. The bases of the trapezoid are the segments of lines

k

and

m

from the

y

-axis to their intersection points with line

l

. The length of the base on line

k

is

9\sqrt{5}

and the length of the base on line

m

is

13\sqrt{5}

(when

a = 25

). Using the formula for the area of a trapezoid,

\text{Area} = \frac{9\sqrt{5} + 13\sqrt{5}}{2} \times 2\sqrt{5} = 11\sqrt{5} \times 2\sqrt{5} = 110

.

Estimated Time:3m 0s

Question 51Question →

A water tank contains 24 gallons of water. Water is being drained from the tank at a constant rate. After 8 minutes, the tank contains 14 gallons of water. If the volume of water in the tank, in gallons, is a linear function of the time in minutes, how many minutes will it take for the tank to contain exactly 4 gallons of water?

Show answer & explanation

Answer: 16

Answer

16

The volume of water in the tank decreases linearly from an initial value of

24

gallons at

t = 0

to

14

gallons at

t = 8

. The constant rate of change (slope) is calculated by dividing the change in volume by the change in time:

\frac{14 - 24}{8 - 0} = -1.25

gallons per minute. Using the slope-intercept form, the volume

V

at time

t

is given by

V = -1.25t + 24

. Setting the volume

V = 4

gives the equation

4 = -1.25t + 24

. Solving for

t

yields

-20 = -1.25t

, which simplifies to

t = 16

.

Step-by-Step Solution

1

Determine the initial state and the state after 8 minutes as coordinate points.

The initial state is

(0, 24)

and the state after

8

minutes is

(8, 14)

.

These coordinates represent the relationship between time and volume of water in the tank.

2

Calculate the slope (constant rate of change) of the linear function.

Slope

m = \frac{14 - 24}{8 - 0} = -1.25

The slope represents the constant rate at which water is being drained from the tank.

3

Formulate the linear equation.

V(t) = -1.25t + 24

Using the slope-intercept form

V(t) = mt + b

, where

b = 24

is the vertical intercept representing the initial volume.

4

Solve for the time

t

when the volume of water is

4

gallons.

4 = -1.25t + 24 \implies -20 = -1.25t \implies t = 16

To find the time at which the volume decreases to exactly

4

gallons.

Key Concept

Linear Functions and Graphs

Question 52Question →

If $\frac{2}{5}(3x - 4) - \frac{1}{3}(2x + 5) = \frac{1}{5}x + \frac{11}{15}$ , what is the value of $2x - 7$ ?

Show answer & explanation

Answer: 17

Answer

The correct answer is 17.

To find the value of

2x - 7

, first solve the linear equation for

x

. Distributing the coefficients on the left side of the equation gives

\frac{6}{5}x - \frac{8}{5} - \frac{2}{3}x - \frac{5}{3} = \frac{1}{5}x + \frac{11}{15}

. Combining the variable terms and constants on the left side results in

\frac{8}{15}x - \frac{49}{15} = \frac{3}{15}x + \frac{11}{15}

. Subtracting

\frac{3}{15}x

and adding

\frac{49}{15}

to both sides yields

\frac{5}{15}x = \frac{60}{15}

, which simplifies to

\frac{1}{3}x = 4

, or

x = 12

. Finally, substituting 12 into the expression

2x - 7

gives

2(12) - 7 = 17

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses.

\frac{6}{5}x - \frac{8}{5} - \frac{2}{3}x - \frac{5}{3} = \frac{1}{5}x + \frac{11}{15}

To eliminate parentheses and allow grouping of like terms.

2

Combine the variable terms and the constant terms on the left side using a common denominator of 15.

\frac{8}{15}x - \frac{49}{15} = \frac{3}{15}x + \frac{11}{15}

To simplify the linear equation into a standard two-sided form.

3

Subtract the variable term from the right side and add the constant term from the left side.

\frac{5}{15}x = \frac{60}{15}

, which simplifies to

x = 12

To isolate the variable

x

on one side of the equation.

4

Evaluate the expression

2x - 7

using the value of

x

.

2(12) - 7 = 17

To solve for the final requested quantity.

Key Concept

Linear Equations in One Variable

Question 53Question →

The table below shows some values of the linear function $f$ .

$x$	$f(x)$
$2$	$11$
$4$	$17$
$6$	$23$

What is the $y$ -coordinate of the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5, representing the y-coordinate of the y-intercept of the graph of f.

To find the

y

-intercept of the linear function, we first determine its slope using the points

(2, 11)

and

(4, 17)

from the table. The slope

m

is

\frac{17 - 11}{4 - 2} = \frac{6}{2} = 3

. Next, we use the slope-intercept equation

f(x) = mx + b

. Substituting

m = 3

and the point

(2, 11)

gives

11 = 3(2) + b

, which simplifies to

11 = 6 + b

. Solving for

b

yields

5

. Therefore, the

y

-coordinate of the

y

-intercept of the graph of

f

is

5

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear function using two coordinate pairs from the table.

The slope is

m = 3

.

A linear function has a constant slope, which can be found using the formula

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

.

2

Substitute the slope and one of the points into the slope-intercept equation

f(x) = mx + b

to solve for the

y

-intercept

b

.

The

y

-intercept

b

is

5

.

The

y

-coordinate of the

y

-intercept of the graph of

y = f(x)

is the value of

b

in the equation

f(x) = mx + b

.

Key Concept

Finding the y-intercept of a linear function from a table of values.

Estimated Time:45s

Question 54Question →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $3x + 4y = 48$ . The graph of $f$ intersects the $x$ -axis at the point $(p, 0)$ and the $y$ -axis at the point $(0, q)$ , where $q > 0$ . If the distance between the points $(p, 0)$ and $(0, q)$ is $15$ , what is the value of $q$ ?

Show answer & explanation

Answer: 12

Answer

12

First, find the slope of the given line by rewriting

3x + 4y = 48

in slope-intercept form:

y = -\frac{3}{4}x + 12

. The slope is

-\frac{3}{4}

. The graph of the linear function

f

is perpendicular to this line, so its slope is the negative reciprocal,

\frac{4}{3}

. With a

y

-intercept of

(0, q)

, the equation of

f

is

y = \frac{4}{3}x + q

. Setting

y = 0

gives the

x

-intercept

(p, 0) = (-\frac{3}{4}q, 0)

. The distance between these intercepts is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

(since

q > 0

). Given that the distance is

15

, we solve

\frac{5}{4}q = 15

to find

q = 12

.

Step-by-Step Solution

1

Find the slope of the line

3x + 4y = 48

.

The slope of the line is

-\frac{3}{4}

.

To find the slope of the perpendicular line

f

, we first need the slope of the given line. Rewriting

3x + 4y = 48

in slope-intercept form gives

y = -\frac{3}{4}x + 12

.

2

Determine the slope of

f

.

The slope of

f

is

\frac{4}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Find the

x

-intercept of

f

in terms of

q

.

The

x

-intercept of

f

is

(-\frac{3}{4}q, 0)

.

Since the

y

-intercept of

f

is

(0, q)

, the equation of

f

is

y = \frac{4}{3}x + q

. Setting

y = 0

gives

0 = \frac{4}{3}x + q

, which simplifies to

x = -\frac{3}{4}q

.

4

Use the distance formula between the intercepts to solve for

q

.

The value of

q

is

12

.

The distance between

(-\frac{3}{4}q, 0)

and

(0, q)

is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

since

q > 0

. Setting this distance to

15

gives

\frac{5}{4}q = 15

, which yields

q = 12

.

Key Concept

Properties of perpendicular lines, finding intercepts, and utilizing the distance formula in coordinate geometry.

Question 55Question →

If $3(2n - 5) = 4n + 9$ , what is the value of $n$ ?

Show answer & explanation

Answer: 12

Answer

12

To solve the equation

3(2n - 5) = 4n + 9

, we first distribute the 3 to the terms inside the parentheses to get

6n - 15 = 4n + 9

. Next, we subtract

4n

from both sides to group the variable terms, giving

2n - 15 = 9

. We then add 15 to both sides to isolate the variable term, resulting in

2n = 24

. Finally, dividing both sides by 2 gives the solution

n = 12

.

Step-by-Step Solution

1

Distribute 3 to the terms inside the parentheses.

6n - 15 = 4n + 9

To simplify the left side of the equation and remove parentheses.

2

Subtract

4n

from both sides of the equation.

2n - 15 = 9

To group the variable terms on one side of the equation.

3

Add 15 to both sides of the equation.

2n = 24

To isolate the term with the variable

n

.

4

Divide both sides of the equation by 2.

n = 12

To solve for

n

.

Key Concept

Solving a linear equation in one variable by applying the distributive property and isolating the variable.

Question 56Question →

Consider the system of equations below.

x = 2y

3x - y = 10

If $(x, y)$ is the solution to the system of equations above, what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is

4

.

The system of equations can be solved by substituting

x = 2y

from the first equation into the second equation, which gives

3(2y) - y = 10

. Simplifying this equation yields

6y - y = 10

, or

5y = 10

. Dividing by

5

gives

y = 2

. Substituting

y = 2

back into the first equation yields

x = 2(2) = 4

. Therefore, the value of

x

is

4

.

Step-by-Step Solution

1

Substitute the expression for

x

from the first equation into the second equation.

3(2y) - y = 10

Since the first equation gives

x

in terms of

y

, substituting it into the second equation reduces the system to a single linear equation in one variable.

2

Solve the resulting equation for

y

.

y = 2

Simplifying

3(2y) - y = 10

yields

6y - y = 10

, which simplifies further to

5y = 10

. Dividing both sides by

5

gives

y = 2

.

3

Substitute the value of

y

back into the first equation to solve for

x

.

x = 4

Using

x = 2y

and substituting

y = 2

gives

x = 2(2) = 4

.

Key Concept

Solving systems of linear equations using the substitution method.

Question 57Question →

In the equation below,

a

and

b

are positive constants.

\frac{2}{3} \left( \frac{3}{2}x - a \right) - \frac{3}{4} \left( 8 - bx \right) = \frac{5}{2}(x - 3) - \frac{1}{2}

If the equation has infinitely many solutions, what is the value of

a^b

?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The correct answer is 9. Expanding the left side of the equation yields

\left(1 + \frac{3}{4}b\right)x - \left(6 + \frac{2}{3}a\right)

, and simplifying the right side yields

\frac{5}{2}x - 8

. For the linear equation to have infinitely many solutions, the coefficient of

x

on the left,

1 + \frac{3}{4}b

, must equal the coefficient of

x

on the right,

\frac{5}{2}

, which gives

b = 2

. Similarly, the constant term on the left,

-\left(6 + \frac{2}{3}a\right)

, must equal the constant term on the right,

-8

, which simplifies to

6 + \frac{2}{3}a = 8

and gives

a = 3

. Evaluating

a^b

with these values yields

3^2 = 9

.

Step-by-Step Solution

1

Expand both sides of the equation to collect like terms.

\left( 1 + \frac{3}{4}b \right)x - \left( 6 + \frac{2}{3}a \right) = \frac{5}{2}x - 8

Expanding allows us to compare the coefficient of

x

and the constant term on each side of the equation.

2

Equate the coefficients of

x

on both sides of the equation.

1 + \frac{3}{4}b = \frac{5}{2} \implies b = 2

For a linear equation to have infinitely many solutions, the coefficient of

x

must be identical on both sides.

3

Equate the constant terms on both sides of the equation.

-\left( 6 + \frac{2}{3}a \right) = -8 \implies a = 3

For a linear equation to have infinitely many solutions, the constant terms must also be identical on both sides.

4

Calculate the value of

a^b

using the solved values of

a

and

b

.

3^2 = 9

The question asks for the value of the expression

a^b

where

a = 3

and

b = 2

.

Key Concept

A linear equation in one variable of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

.

Alternative Method

Since the equation must hold for all values of

x

if it has infinitely many solutions, you can substitute convenient values for

x

to solve for

a

and

b

directly. Substituting

x = 0

simplifies the equation to

-\frac{2}{3}a - 6 = -8

, which quickly yields

a = 3

. Then, substituting

x = 2

and

a = 3

simplifies the equation to

-\frac{3}{4}(8 - 2b) = -3

, which yields

b = 2

. Calculating

a^b

gives

3^2 = 9

.

Estimated Time:3m 0s

Question 58Question →

If $\frac{5}{3}(3x - 6) - \frac{1}{2}(4x + 8) = 2$ , what is the value of $3x - 5$ ?

Show answer & explanation

Answer: 11

Answer

11

Distributing the fraction

\frac{5}{3}

to

(3x - 6)

yields

5x - 10

. Distributing the term

-\frac{1}{2}

to

(4x + 8)

yields

-2x - 4

. Combining these results gives

(5x - 2x) + (-10 - 4) = 2

, which simplifies to

3x - 14 = 2

. Adding 14 to both sides yields

3x = 16

. Finally, subtracting 5 from both sides of this equation gives

3x - 5 = 11

.

Step-by-Step Solution

1

Distribute

\frac{5}{3}

and

-\frac{1}{2}

to the terms within their respective parentheses.

5x - 10 - 2x - 4 = 2

This removes the parentheses and allows us to group terms.

2

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

3x - 14 = 2

Simplifying the expression on the left side makes it easier to solve.

3

Add 14 to both sides of the equation to isolate the term

3x

.

3x = 16

Since the question asks for the value of

3x - 5

, isolating

3x

allows direct evaluation.

4

Subtract 5 from both sides of the equation

3x = 16

.

3x - 5 = 11

This directly yields the required value without needing to compute the fractional value of

x

first.

Key Concept

Solving linear equations in one variable by distribution, combining like terms, and evaluating expressions.

Estimated Time:1m 30s

Question 59Question →

In the $xy$ -plane, the graph of a linear function $f$ has a positive $y$ -intercept and a positive $x$ -intercept. The area of the triangular region in the first quadrant bounded by the graph of $f$ and the coordinate axes is $36$ . If the graph of $f$ passes through the point $(4, 4)$ and has a slope less than $-1$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 12

Answer

The

y

-coordinate of the

y

-intercept of the graph of

f

is

12

.

The correct answer is

12

because the system of equations derived from the area constraint (

qr = 72

) and the point constraint (

q+r=18

) yields two possible values for the

y

-intercept:

6

or

12

. The condition that the slope must be less than

-1

means that the line must be steeper than a slope of

-1

, which requires the

y

-intercept to be larger than the

x

-intercept (

r > q

). Thus, the

y

-intercept is

12

.

Step-by-Step Solution

1

Express the equation of the line using intercept form.

\frac{x}{q} + \frac{y}{r} = 1

, where

q > 0

is the

x

-intercept and

r > 0

is the

y

-intercept.

Since the line intersects the positive axes, this form directly relates the intercepts to the coordinates of points on the line.

2

Use the area of the triangle to find a relationship between

q

and

r

.

qr = 72

The area of the right triangle formed by the axes and the intercepts is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}qr = 36

.

3

Substitute the point

(4, 4)

into the equation of the line and simplify using the area relationship.

q + r = 18

Plugging in

x = 4

and

y = 4

yields

\frac{4}{q} + \frac{4}{r} = 1 \implies 4(q + r) = qr

. Substituting

qr = 72

gives

4(q + r) = 72 \implies q + r = 18

.

4

Solve the system of equations

q + r = 18

and

qr = 72

.

(q, r) = (6, 12)

or

(q, r) = (12, 6)

Substituting

r = 18 - q

into

qr = 72

yields

q(18 - q) = 72 \implies q^2 - 18q + 72 = 0 \implies (q - 6)(q - 12) = 0

.

5

Apply the slope condition to determine the unique value of

r

.

r = 12

The slope of the line is

m = -\frac{r}{q}

. If

q = 12

and

r = 6

, then

m = -0.5

, which is not less than

-1

. If

q = 6

and

r = 12

, then

m = -2

, which is less than

-1

. Thus,

r = 12

is the correct

y

-intercept.

Key Concept

Using intercepts and area to determine the equation of a linear function under constraints.

Question 60Question →

The table below shows some values of $x$ and the corresponding values of $f(x)$ for a linear function $f$ .

$x$	$f(x)$
3	7
6	12
9	17

What is the value of $f(15)$ ?

Show answer & explanation

Answer: 27

Answer

The value of the linear function evaluated at 15 is 27.

To find

f(15)

for the linear function, we determine its constant rate of change. Using points

(3, 7)

and

(6, 12)

, the change in

f(x)

is

12 - 7 = 5

for a change in

x

of

6 - 3 = 3

. This gives a slope of

\frac{5}{3}

. Using point-slope form with

(3, 7)

gives the equation

f(x) - 7 = \frac{5}{3}(x - 3)

, which simplifies to

f(x) = \frac{5}{3}x + 2

. Substituting

x = 15

yields

f(15) = \frac{5}{3}(15) + 2 = 25 + 2 = 27

. Alternatively, since

x

increases by 6 from 9 to 15, which is twice the step size of 3, the value of

f(x)

must increase by

2 \times 5 = 10

from

17

, resulting in

17 + 10 = 27

.

Step-by-Step Solution

1

Calculate the slope of the linear function

slope

m = \frac{5}{3}

Linear functions have a constant rate of change, which is the slope.

2

Determine the equation of the function

f(x) = \frac{5}{3}x + 2

Using the slope and one point from the table helps define the function for all inputs.

3

Evaluate the function at

x = 15

f(15) = 27

Substitute the given input value into the function equation to find the corresponding output value.

Key Concept

Linear Functions and Graphs

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept