Algebra Practice Questions — SAT — Page 2

Consider the system of equations below.

y = 3x - 4

x + y = 8

What is the value of $x$ ?

3

5

1

2

Show answer & explanation

Answer: $3$

Answer

The value of

x

is

3

.

To solve the system, substitute the value of

y

from the first equation into the second equation:

x + (3x - 4) = 8

. Combining the

x

terms gives

4x - 4 = 8

. Adding

4

to both sides of the equation results in

4x = 12

. Dividing both sides by

4

isolates the variable, giving

x = 3

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

x + (3x - 4) = 8

This eliminates the variable

y

to create a single equation in terms of

x

.

2

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

4x - 4 = 8

Grouping the

x

variables simplifies the expression.

3

Add

4

to both sides of the equation to isolate the term with

x

.

4x = 12

Performing the inverse operation of subtraction isolates the variable term.

4

Divide both sides of the equation by

4

to solve for

x

.

x = 3

Performing the inverse operation of multiplication isolates the variable

x

.

Key Concept

Solving a system of linear equations using the substitution method.

Question 22Question →

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

y = 2x + 1

3x + 2y = 16

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Substituting

y = 2x + 1

into the second equation yields

3x + 2(2x + 1) = 16

. Expanding this expression gives

3x + 4x + 2 = 16

, which simplifies to

7x + 2 = 16

. Subtracting 2 from both sides gives

7x = 14

, and dividing by 7 yields

x = 2

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

3x + 2(2x + 1) = 16

This reduces the system of two variables to a single equation in terms of

x

.

2

Expand and simplify the equation by distributing and combining like terms.

7x + 2 = 16

Simplifying the equation makes it easier to isolate the variable

x

.

3

Isolate the variable

x

by performing inverse operations.

x = 2

Subtracting 2 and then dividing by 7 isolates

x

to find its value.

Key Concept

Solving a system of linear equations using the substitution method.

Alternative Method

Alternatively, you can multiply the first equation by 2 to get

-4x + 2y = 2

and subtract it from

3x + 2y = 16

to eliminate

y

, which also yields

7x = 14

and thus

x = 2

.

Estimated Time:1m 0s

Question 23Question →

In the system of linear equations shown below, what value of $x$ satisfies both equations?

y = 2x - 5

3x - 2y = 12

-22

-2

2

-9

Show answer & explanation

Answer: $-2$

Answer

-2

To find the value of

x

, substitute the expression for

y

from the first equation,

y = 2x - 5

, into the second equation,

3x - 2y = 12

. This yields

3x - 2(2x - 5) = 12

. Distributing the

-2

across the terms in the parentheses results in

3x - 4x + 10 = 12

. Simplifying the left side of the equation by combining like terms gives

-x + 10 = 12

. Subtracting

10

from both sides results in

-x = 2

. Finally, multiplying or dividing both sides by

-1

yields

x = -2

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation

3x - 2(2x - 5) = 12

To eliminate

y

and obtain an equation containing only

x

.

2

Distribute

-2

to the terms inside the parentheses

3x - 4x + 10 = 12

To remove parentheses and prepare the equation for simplifying.

3

Combine like terms and solve for

x

-x + 10 = 12

, which gives

-x = 2

, and thus

x = -2

To isolate and find the value of

x

.

Key Concept

Solving systems of linear equations using substitution

Question 24Question →

A line representing the linear function $f$ is graphed in the $xy$ -plane. The line intersects the $y$ -axis at $(0, 3)$ and also passes through the point $(2, 11)$ . What is the slope of this line?

Show answer & explanation

Answer: 4

Answer

The slope of the line is 4.

The slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Substituting the given points

(0, 3)

and

(2, 11)

into the formula gives

m = \frac{11 - 3}{2 - 0} = \frac{8}{2} = 4

.

Step-by-Step Solution

1

Identify two coordinates on the line from the given information.

The points are

(0, 3)

and

(2, 11)

.

To find the slope of a line, we need at least two points on that line.

2

Apply the slope formula

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

.

m = \frac{11 - 3}{2 - 0}

The slope formula calculates the ratio of the vertical change (rise) to the horizontal change (run).

3

Perform the subtraction and division to find the slope value.

m = \frac{8}{2} = 4

Simplifying the fraction gives the final numerical slope.

Key Concept

Calculating the slope of a linear function given two points

Question 25Question →

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 26Question →

In the equation below,

k

is a constant.

5x - 2(3 - k)x = 9x - 1

If the equation has no solution, what is the value of

k

?

-5

0

5

4

Show answer & explanation

Answer: 5

Answer

5

To find the value of

k

for which the equation has no solution, we simplify the left side of the equation:

5x - 2(3 - k)x = 5x - 6x + 2kx = -x + 2kx = (2k - 1)x

. This yields the simplified equation

(2k - 1)x = 9x - 1

. A linear equation in one variable has no solution when the variable terms on both sides cancel each other out (meaning the coefficients of

x

are equal) but the constant terms are different. Setting the coefficients of

x

equal to each other gives

2k - 1 = 9

. Adding

1

to both sides gives

2k = 10

, and dividing by

2

yields

k = 5

. Since the constant term on the left side is

0

and the constant term on the right side is

-1

, the constants are different, confirming that the equation has no solution when

k

equals

5

.

Step-by-Step Solution

1

Distribute the term

-2

to the expressions inside the parentheses on the left side of the equation.

5x - 6x + 2kx = 9x - 1

To eliminate the parentheses so that all like terms can be grouped.

2

Combine the

x

terms on the left side of the equation.

-x + 2kx = 9x - 1

, which can be factored as

(2k - 1)x = 9x - 1

Grouping the coefficients of the variable

x

allows direct comparison of both sides of the linear equation.

3

Set the coefficient of

x

on the left side equal to the coefficient of

x

on the right side.

2k - 1 = 9

For a linear equation in one variable to have no solution, the variable terms on both sides must cancel out (meaning their coefficients must be equal), while the constant terms must remain unequal.

4

Solve the resulting equation for

k

.

2k = 10 \implies k = 5

Isolating

k

by adding

1

to both sides and then dividing by

2

determines the specific constant value.

5

Verify that the constant terms are different when

k = 5

.

Substitute

k = 5

back into the original equation to get

9x = 9x - 1

, which simplifies to

0 = -1

.

Since

0 = -1

is a false statement, the equation has no solution, confirming that

k = 5

is correct.

Key Concept

Linear Equations in One Variable

Alternative Method

Instead of algebraic simplification, the value of

k

can be found by substituting the answer choices into the equation to see which value eliminates the variable

x

while leaving an untrue statement. Plugging in

5

for

k

gives

5x - 2(3 - 5)x = 9x - 1 \implies 5x - 2(-2)x = 9x - 1 \implies 9x = 9x - 1 \implies 0 = -1

. Because this statement is false, the equation has no solution, verifying that

5

is the correct answer.

Estimated Time:2m 0s

Question 27Question →

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 28Question →

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(0, 1)$ and $(2, k)$ . The graph of another linear function $g$ is perpendicular to the graph of $f$ and passes through the points $(-4, 8)$ and $(2, k)$ . Which of the following could be the value of $k$ ?

2

8

5

9

Show answer & explanation

Answer: 5

Answer

5

The slope of the line representing function

f

is calculated as

m_f = \frac{k - 1}{2}

, and the slope of the line representing function

g

is

m_g = \frac{k - 8}{6}

. Since the lines are perpendicular, the product of their slopes must be

-1

. This gives the equation

\frac{k - 1}{2} \cdot \frac{k - 8}{6} = -1

, which simplifies to

(k - 1)(k - 8) = -12

. Expanding and setting the quadratic equation to zero gives

k^2 - 9k + 20 = 0

. Factoring this expression yields

(k - 4)(k - 5) = 0

. Thus,

k

can be either

4

or

5

. Since 5 is the only option listed, the option containing 5 is the correct answer.

Step-by-Step Solution

1

Express the slope of the linear function

f

in terms of

k

.

m_f = \frac{k - 1}{2 - 0} = \frac{k - 1}{2}

The slope formula is

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

.

2

Express the slope of the linear function

g

in terms of

k

.

m_g = \frac{k - 8}{2 - (-4)} = \frac{k - 8}{6}

The slope formula is applied to the points

(-4, 8)

and

(2, k)

.

3

Apply the perpendicular lines condition to set up an equation for

k

.

m_f \cdot m_g = -1 \implies \left(\frac{k - 1}{2}\right)\left(\frac{k - 8}{6}\right) = -1

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

.

4

Solve the quadratic equation for

k

.

(k - 1)(k - 8) = -12 \implies k^2 - 9k + 8 = -12 \implies k^2 - 9k + 20 = 0 \implies (k - 4)(k - 5) = 0

Multiply both sides by 12, expand the product, move all terms to one side, and factor the quadratic expression.

5

Identify the possible values of

k

and match with the options.

k = 4

or

k = 5

. Since 5 is the only value present in the options, the correct value is 5.

Both values satisfy the condition, but only one is listed among the multiple-choice options.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (their product is

-1

).

Question 29Question →

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 30Question →

If $20 - 2(x + 3) = 8$ , what is the value of $x$ ?

-11

-3

3

9

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. Distributing

-2

to the terms inside the parentheses gives

20 - 2x - 6 = 8

. Simplifying the left side yields

14 - 2x = 8

. Subtracting 14 from both sides results in

-2x = -6

. Dividing both sides by

-2

gives

x = 3

.

Step-by-Step Solution

1

Distribute the coefficient

-2

to all terms inside the parentheses

(x + 3)

.

20 - 2x - 6 = 8

Applying the distributive property requires multiplying both

x

and

3

by

-2

.

2

Combine the constant terms

20

and

-6

on the left side of the equation.

14 - 2x = 8

Simplifying the expression makes it easier to isolate the variable term.

3

Subtract

14

from both sides of the equation to isolate the variable term

-2x

.

-2x = -6

This moves the constant term to the opposite side of the equation.

4

Divide both sides of the equation by

-2

to solve for

x

.

x = 3

Dividing by the coefficient of the variable isolates the variable completely.

Key Concept

Solving linear equations in one variable with parentheses by distributing coefficients, combining like terms, and isolating the variable.

Question 31Question →

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 32Question →

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 33Question →

If $\frac{2}{3}(x - 6) - \frac{1}{2}(x - 4) = 4$ , what is the value of $3x - 12$ ?

168

96

36

24

Show answer & explanation

Answer: 96

Answer

96

The correct answer is 96. Multiplying the entire equation by the common denominator 6 clears the fractions to yield

4(x - 6) - 3(x - 4) = 24

. Distributing the factors gives

4x - 24 - 3x + 12 = 24

, which simplifies to

x - 12 = 24

. Adding 12 to both sides determines that

x = 36

. Evaluating the expression

3x - 12

with

x = 36

yields

3(36) - 12 = 96

.

Step-by-Step Solution

1

Multiply the entire equation by 6 (the least common multiple of the denominators 3 and 2) to eliminate the fractions.

4(x - 6) - 3(x - 4) = 24

Clearing the denominators simplifies the algebraic manipulation and reduces the risk of fractional arithmetic errors.

2

Distribute the coefficients to the terms inside the parentheses, taking care to distribute the negative sign for the second term.

4x - 24 - 3x + 12 = 24

Applying the distributive property expands the expression so that like terms can be grouped.

3

Combine like terms on the left side of the equation.

x - 12 = 24

Grouping

4x - 3x

to get

x

and

-24 + 12

to get

-12

simplifies the equation to a single variable and constant.

4

Isolate the variable

x

by adding 12 to both sides of the equation.

x = 36

Adding the opposite of the constant term isolates the variable on one side.

5

Substitute

x = 36

into the requested expression

3x - 12

to find its value.

3(36) - 12 = 96

The question asks for the value of the expression

3x - 12

, not just the value of

x

.

Key Concept

Solving multi-step linear equations in one variable and evaluating algebraic expressions.

Estimated Time:1m 30s

Question 34Question →

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 35Question →

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 36Question →

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 37Question →

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

$x$	$g(x)$
$2$	$5$
$4$	$11$
$6$	$17$

What is the slope of the graph of $g$ in the $xy$ -plane?

\frac{1}{3}

6

3

\frac{5}{2}

Show answer & explanation

Answer: $3$

Answer

The slope of the graph is

3

.

The slope of a linear function is the constant rate of change, calculated as the change in the function values divided by the change in the input values. Choosing the points

(2, 5)

and

(4, 11)

, the slope is

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

. This same slope is obtained using any other pair of points from the table, such as

(4, 11)

and

(6, 17)

, where the slope is

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

.

Step-by-Step Solution

1

Select two points from the table to find the change in the input and output values.

Using the points

(2, 5)

and

(4, 11)

, the change in the output

g(x)

is

11 - 5 = 6

, and the change in the input

x

is

4 - 2 = 2

.

The slope is defined as the change in the vertical coordinate divided by the change in the horizontal coordinate.

2

Divide the change in

g(x)

by the change in

x

to find the slope.

The slope is

\frac{6}{2} = 3

.

Dividing the vertical change by the horizontal change yields the constant rate of change of the linear function.

Key Concept

Finding the slope of a linear function from a table of values

Estimated Time:45s

Question 38Question →

The graph of a linear function is shown in the $xy$ -plane. The line passes through the points $(-2, -3)$ and $(2, 5)$ . If the point $(k, 11)$ lies on the line, what is the value of $k$ ?

9

14

5

6

Show answer & explanation

Answer: 5

Answer

5

The slope of the line is calculated as

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

. Using the point-slope form with the point

(2, 5)

, the equation of the line is

y - 5 = 2(x - 2)

, which simplifies to

y = 2x + 1

. Substituting the point

(k, 11)

into this equation gives

11 = 2k + 1

, which solves to

k = 5

.

Step-by-Step Solution

1

Calculate the slope of the line using the points

(-2, -3)

and

(2, 5)

m = 2

The slope formula is

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

. Substituting the given coordinates:

m = \frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2

.

2

Determine the equation of the line using point-slope form

y = 2x + 1

Using

y - y_1 = m(x - x_1)

with the point

(2, 5)

and

m = 2

, we get

y - 5 = 2(x - 2)

, which simplifies to

y = 2x + 1

.

3

Substitute

(k, 11)

into the line equation and solve for

k

k = 5

Substituting

x = k

and

y = 11

yields

11 = 2k + 1

. Subtracting 1 from both sides gives

10 = 2k

, and dividing by 2 gives

k = 5

.

Key Concept

Finding the linear equation from coordinates and solving for variables

Question 39Question →

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 40Question →

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

Algebra

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

Alternative Method

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Common Mistakes

Alternative Method

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Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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