Algebra Practice Questions — SAT — Page 7

A linear relationship between $x$ and $y$ is defined by the values in the table below.

$x$	$y$
2	7
4	13
6	19

Which of the following equations represents this relationship?

y = 3x - 5

y = \frac{1}{3}x + \frac{19}{3}

y = 3x + 1

y = 6x - 5

Show answer & explanation

Answer: $y = 3x + 1$

Answer

The equation representing the relationship is

y = 3x + 1

.

The correct equation is

y = 3x + 1

. The slope

m

of the linear function can be determined from the table values

(2, 7)

and

(4, 13)

as

\frac{13 - 7}{4 - 2} = \frac{6}{2} = 3

. Substituting the slope

m = 3

and the point

(2, 7)

into the slope-intercept form

y = mx + b

gives

7 = 3(2) + b

, which simplifies to

7 = 6 + b

, so

b = 1

. Therefore, the equation is

y = 3x + 1

.

Step-by-Step Solution

1

Find the slope (

m

) of the linear relationship using the points

(2, 7)

and

(4, 13)

from the table.

m = 3

The slope is the ratio of the change in

y

to the change in

x

between any two points on the line:

m = \frac{13 - 7}{4 - 2} = \frac{6}{2} = 3

.

2

Use the slope

m = 3

and the point

(2, 7)

to find the

y

-intercept (

b

) of the line.

b = 1

Substituting the coordinates and slope into the slope-intercept equation

y = mx + b

gives

7 = 3(2) + b

, which simplifies to

7 = 6 + b

, leading to

b = 1

.

3

Substitute the slope and

y

-intercept into the slope-intercept form.

y = 3x + 1

Combining the slope of

3

and

y

-intercept of

1

yields the equation

y = 3x + 1

.

Key Concept

Determining a linear equation from a table of values

Estimated Time:45s

Question 122Question →

In the system of equations below, $k$ is a constant.

kx - 3y = 4

(k-2)x - 5y = 8

If the system has no solution, what is the value of $k$ ?

-1

5

-3

3

Show answer & explanation

Answer: -3

Answer

-3

To find the value of

k

for which the system of linear equations has no solution, we determine when the two lines represented by the equations are parallel and distinct. Parallel lines have equal slopes. We can express each equation in slope-intercept form,

y = mx + b

. For the first equation,

kx - 3y = 4

simplifies to

y = \frac{k}{3}x - \frac{4}{3}

, giving a slope of

\frac{k}{3}

. For the second equation,

(k-2)x - 5y = 8

simplifies to

y = \frac{k-2}{5}x - \frac{8}{5}

, giving a slope of

\frac{k-2}{5}

. Equating the two slopes yields

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

. Cross-multiplying gives

5k = 3(k-2)

, which simplifies to

5k = 3k - 6

. Subtracting

3k

from both sides gives

2k = -6

, and dividing by

2

results in

k = -3

. Since the y-intercepts

-\frac{4}{3}

and

-\frac{8}{5}

are distinct, the lines are parallel and do not intersect, meaning there is no solution.

Step-by-Step Solution

1

Express both equations in slope-intercept form (

y = mx + b

) to find their slopes.

For the first equation,

kx - 3y = 4 \implies 3y = kx - 4 \implies y = \frac{k}{3}x - \frac{4}{3}

(slope is

\frac{k}{3}

). For the second equation,

(k-2)x - 5y = 8 \implies 5y = (k-2)x - 8 \implies y = \frac{k-2}{5}x - \frac{8}{5}

(slope is

\frac{k-2}{5}

).

A system of two linear equations has no solution if the lines are parallel (slopes are equal) and their y-intercepts are different.

2

Set the slopes equal to each other.

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

Since the lines must be parallel, their slopes must be equivalent.

3

Solve for

k

by cross-multiplying.

5k = 3(k-2) \implies 5k = 3k - 6 \implies 2k = -6 \implies k = -3

.

Cross-multiplication isolates the variable

k

. Since the y-intercepts (

-\frac{4}{3}

and

-\frac{8}{5}

) are different, this value of

k

guarantees the lines are parallel and distinct.

Key Concept

Determining parameters for a system of linear equations with no solution

Alternative Method

Alternatively, align the coefficients of

y

to eliminate the

y

terms. Multiply the first equation by

5

and the second equation by

-3

:

5(kx - 3y) = 5(4) \implies 5kx - 15y = 20

-3((k-2)x - 5y) = -3(8) \implies -3(k-2)x + 15y = -24

Adding these two equations yields:

(5k - 3(k-2))x = -4

(2k + 6)x = -4

For the system to have no solution, the coefficient of

x

must be

0

while the constant on the right side is non-zero (which it is,

-4

). Setting the coefficient of

x

to

0

:

2k + 6 = 0 \implies k = -3

Estimated Time:2m 0s

Question 123Question →

A line in the $xy$ -plane is represented by the equation $3x - 4y = 24$ . What is the $y$ -coordinate of the $y$ -intercept of this line?

-6

6

8

\frac{3}{4}

Show answer & explanation

Answer: -6

Answer

The

y

-coordinate of the

y

-intercept is

-6

.

The correct answer is

-6

. The

y

-intercept of a line is the point where it intersects the

y

-axis, which occurs when the

x

-coordinate is

0

. Substituting

x = 0

into the given equation

3x - 4y = 24

yields

-4y = 24

. Dividing both sides by

-4

gives

y = -6

.

Step-by-Step Solution

1

Substitute

x = 0

into the equation to find the

y

-intercept.

3(0) - 4y = 24

By definition, the

y

-intercept of a line in the coordinate plane is the point where the line crosses the

y

-axis, which always has an

x

-coordinate of

0

.

2

Simplify the equation.

-4y = 24

Multiplying

3

by

0

yields

0

, leaving only the term containing

y

on the left side of the equation.

3

Solve for

y

by isolating the variable.

y = -6

Dividing both sides of the simplified equation by

-4

isolates

y

and gives the final coordinate value.

Key Concept

Finding the y-intercept of a linear equation in two variables by setting x = 0.

Estimated Time:45s

Question 124Question →

For what value of the constant $k$ does the equation $\frac{1}{2}(2kx - 6) - \frac{2}{3}(9 - 3x) = 5x + 4$ have no solution for $x$ ?

3

7

1.5

-7

Show answer & explanation

Answer: 3

Answer

The correct value of

k

is

3

.

To find the value of

k

for which the equation has no solution, we first distribute the coefficients on the left side of the equation to get

kx - 3 - 6 + 2x = 5x + 4

. Combining the like terms on the left side gives

(k + 2)x - 9 = 5x + 4

. Moving all the

x

terms to one side yields

(k - 3)x = 13

. For a linear equation of the form

Ax = B

to have no solution, the coefficient of the variable

A

must equal

0

while the constant

B

must be non-zero. Setting

k - 3 = 0

gives the correct value of

3

.

Step-by-Step Solution

1

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

The term

\frac{1}{2}(2kx - 6)

simplifies to

kx - 3

, and the term

-\frac{2}{3}(9 - 3x)

simplifies to

-6 + 2x

.

Applying the distributive property simplifies the parenthetical expressions.

2

Substitute the simplified expressions back into the equation and group like terms.

The equation becomes

kx - 3 - 6 + 2x = 5x + 4

, which simplifies to

(k + 2)x - 9 = 5x + 4

.

Grouping terms allows us to isolate the variable.

3

Isolate the terms containing

x

on one side of the equation.

Subtract

5x

and add

9

to both sides, yielding

(k - 3)x = 13

.

This puts the equation into the standard linear form

Ax = B

.

4

Set the coefficient of

x

to

0

to find the value of

k

that yields no solution.

Setting

k - 3 = 0

gives

k = 3

. Since

13 \neq 0

, the equation

0x = 13

has no solution.

A linear equation of the form

Ax = B

has no solution if and only if

A = 0

and

B \neq 0

.

Key Concept

Determining the parameter value for which a linear equation in one variable has no solution.

Estimated Time:2m 0s

Question 125Question →

\begin{aligned} 3x - 5y &= 8 \\ kx + 10y &= -3 \end{aligned}

In the system of equations above, $k$ is a constant. If the system has no solution, what is the value of $k$ ?

-2

-6

-1.5

6

Show answer & explanation

Answer: -6

Answer

-6

The correct answer is

-6

. A system of two linear equations has no solution when the equations represent parallel lines, meaning they have the same slope but different

y

-intercepts. Writing the first equation in slope-intercept form gives

y = \frac{3}{5}x - \frac{8}{5}

, so its slope is

\frac{3}{5}

. Writing the second equation in slope-intercept form gives

y = -\frac{k}{10}x - \frac{3}{10}

, so its slope is

-\frac{k}{10}

. Since the

y

-intercepts are different (

-\frac{8}{5} \neq -\frac{3}{10}

), the system will have no solution when their slopes are equal:

\frac{3}{5} = -\frac{k}{10}

. Solving this equation for

k

yields

k = -6

.

Step-by-Step Solution

1

Convert the first equation to slope-intercept form (

y = mx + b

) to find its slope.

y = \frac{3}{5}x - \frac{8}{5}

, so the slope is

\frac{3}{5}

and the

y

-intercept is

-\frac{8}{5}

.

Expressing the line in slope-intercept form directly reveals its slope and

y

-intercept.

2

Convert the second equation to slope-intercept form to express its slope in terms of

k

.

10y = -kx - 3 \Rightarrow y = -\frac{k}{10}x - \frac{3}{10}

, so the slope is

-\frac{k}{10}

and the

y

-intercept is

-\frac{3}{10}

.

This allows comparison of the slope and

y

-intercept of the second line with the first line.

3

Set the slopes equal to each other and solve for

k

.

\frac{3}{5} = -\frac{k}{10} \Rightarrow 30 = -5k \Rightarrow k = -6

.

A system of two linear equations has no solution if and only if the lines are parallel (equal slopes) and have different

y

-intercepts (which

-\frac{8}{5}

and

-\frac{3}{10}

are).

Key Concept

Determining the number of solutions to a system of linear equations based on slope and intercept relationships.

Question 126Question →

In the $xy$ -plane, line $L_1$ is defined by the equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero constants. Line $L_2$ is perpendicular to line $L_1$ and has a $y$ -intercept that is twice the $y$ -intercept of line $L_1$ . If the $x$ -intercept of line $L_2$ is $d$ , which of the following expressions represents $d$ in terms of $a$ , $b$ , and $c$ ?

-\frac{2ac}{b^2}

\frac{2ac}{b^2}

-\frac{2bc}{a^2}

-\frac{2c}{a}

Show answer & explanation

Answer: $-\frac{2ac}{b^2}$

Answer

The expression that represents

d

in terms of

a

,

b

, and

c

is

-\frac{2ac}{b^2}

.

The correct answer is found by first rewriting the equation of the first line in slope-intercept form to determine its slope,

-\frac{a}{b}

, and its

y

-intercept,

\frac{c}{b}

. The second line is perpendicular to the first, so its slope is the negative reciprocal,

\frac{b}{a}

. Its

y

-intercept is twice that of the first line, which is

\frac{2c}{b}

. Setting

y = 0

in the equation of the second line,

y = \frac{b}{a}x + \frac{2c}{b}

, and solving for

x

yields the

x

-intercept

d = -\frac{2ac}{b^2}

.

Step-by-Step Solution

1

Rewrite the equation of line

L_1

in slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

To identify the slope and

y

-intercept of the first line.

2

Determine the slope and

y

-intercept of line

L_2

.

Slope of

L_2

is

\frac{b}{a}

;

y

-intercept of

L_2

is

\frac{2c}{b}

.

Line

L_2

is perpendicular to

L_1

(so its slope is the negative reciprocal of

-\frac{a}{b}

) and its

y

-intercept is twice that of

L_1

.

3

Write the equation of line

L_2

.

y = \frac{b}{a}x + \frac{2c}{b}

Using the slope-intercept form

y = mx + k

for the second line.

4

Find the

x

-intercept

d

of line

L_2

by setting

y = 0

.

d = -\frac{2ac}{b^2}

The

x

-intercept is the value of

x

when

y = 0

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Instead of converting to slope-intercept form, one can use the general form relationships. For

ax + by = c

, the slope is

-a/b

and the

y

-intercept is

c/b

. A perpendicular line has slope

b/a

. With double the

y

-intercept, its equation can be written as

y - \frac{2c}{b} = \frac{b}{a}x

. Setting

y=0

directly gives

-\frac{2c}{b} = \frac{b}{a}d \implies d = -\frac{2ac}{b^2}

.

Estimated Time:2m 0s

Question 127Question →

In the $xy$ -plane, the graph of the linear function $h$ has a slope of $3$ and passes through the point $(5, 8)$ . What is the value of $h(11)$ ?

Show answer & explanation

Answer: 26

Answer

The value of

h(11)

is 26.

The correct answer is 26. Since the function

h

is linear, its rate of change (slope) is constant. A slope of 3 means that for every 1-unit increase in

x

, the value of

h(x)

increases by 3. The change in

x

from 5 to 11 is

11 - 5 = 6

. Therefore, the corresponding increase in

h(x)

is

3 \times 6 = 18

. Adding this increase to the known value of

h(5) = 8

gives

h(11) = 8 + 18 = 26

.

Step-by-Step Solution

1

Determine the change in the input variable

x

from the known point to the target point.

The change in

x

is

11 - 5 = 6

.

This identifies how many units the input increases from

x = 5

to

x = 11

.

2

Multiply the change in

x

by the slope to find the corresponding change in the function value.

The change in the function value is

3 \times 6 = 18

.

The slope represents the constant rate of change (change in

y

divided by change in

x

). Therefore, multiplying the slope by the change in

x

gives the total change in the output.

3

Add the calculated change in the function value to the initial function value.

h(11) = 8 + 18 = 26

.

Adding the total increase in output to the starting output value at

x = 5

yields the value of the function at

x = 11

.

Key Concept

Linear functions have a constant rate of change, which is represented by the slope. The change in the output value is equal to the slope multiplied by the change in the input value.

Alternative Method

Alternatively, find the equation of the line using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the slope

m = 3

and the point

(5, 8)

gives

y - 8 = 3(x - 5)

, which simplifies to

y = 3x - 7

. Thus, the function is defined by

h(x) = 3x - 7

. Evaluating this function at

x = 11

yields

h(11) = 3(11) - 7 = 33 - 7 = 26

.

Estimated Time:1m 0s

Question 128Question →

Which of the following represents all possible values of $x$ that satisfy the inequality $-4(x + 3) \leq 8$ ?

x \geq -5

x \leq -5

x \geq 1

x \geq -4

Show answer & explanation

Answer: $x \geq -5$

Answer

x \geq -5

To solve

-4(x + 3) \leq 8

, first distribute the

-4

to obtain

-4x - 12 \leq 8

. Next, add

12

to both sides to isolate the variable term, resulting in

-4x \leq 20

. Finally, divide both sides by

-4

and reverse the inequality sign because of division by a negative number, which yields

x \geq -5

.

Step-by-Step Solution

1

Distribute the coefficient outside the parentheses to the terms inside.

-4x - 12 \leq 8

Multiplying

-4

by

x

and

-4

by

3

simplifies the left side of the inequality.

2

Isolate the variable term by adding

12

to both sides of the inequality.

-4x \leq 20

Adding

12

eliminates the constant term on the left side:

8 + 12 = 20

.

3

Divide both sides of the inequality by

-4

and reverse the inequality sign.

x \geq -5

Dividing by a negative number reverses the direction of the inequality symbol:

20 / -4 = -5

.

Key Concept

Solving linear inequalities in one variable using the distributive property and division by a negative number.

Estimated Time:45s

Question 129Question →

In the system of equations below, $a$ is a constant.

\begin{aligned} ax + 3y &= 5 \\ 2x - y &= 1 \end{aligned}

If the system has a solution $(x, y)$ that lies in Quadrant IV of the $xy$ -plane, which of the following could be the value of $a$ ?

12

8

-4

-8

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. For the solution

(x, y)

to lie in Quadrant IV, the

x

-coordinate must be positive and the

y

-coordinate must be negative. Isolating

y

in the second equation gives

y = 2x - 1

. Substituting this into the first equation yields

x = \frac{8}{a+6}

. For

x

to be positive, we must have

a > -6

. Substituting

x

back into the expression for

y

gives

y = \frac{10-a}{a+6}

. For

y

to be negative when

a > -6

, we must have

10 - a < 0

, which simplifies to

a > 10

. Among the choices, only 12 is greater than 10.

Step-by-Step Solution

1

Identify the signs of coordinates in Quadrant IV.

x > 0

and

y < 0

In Quadrant IV of the coordinate plane, the horizontal coordinate is positive and the vertical coordinate is negative.

2

Express

y

in terms of

x

from the second equation.

y = 2x - 1

This facilitates substituting

y

into the first equation to solve for

x

.

3

Substitute

y = 2x - 1

into the first equation and solve for

x

.

x = \frac{8}{a + 6}

To find the coordinates of the intersection point in terms of the parameter

a

.

4

Substitute

x = \frac{8}{a + 6}

back into the expression for

y

.

y = \frac{10 - a}{a + 6}

To express the

y

-coordinate of the intersection point in terms of the parameter

a

.

5

Apply the conditions

x > 0

and

y < 0

to solve for

a

.

a > 10

For

x > 0

, we need

a + 6 > 0 \implies a > -6

. For

y < 0

given

a + 6 > 0

, we need

10 - a < 0 \implies a > 10

.

6

Evaluate the options to find a value of

a

that satisfies the condition.

12

Among the options (12, 8, -4, -8), only 12 is greater than 10.

Key Concept

Solving systems of linear equations with parameters and applying quadrant constraints

Question 130Question →

A gym membership costs $\$35$ per month plus an additional $\$5$ per fitness class attended. If a member wants to spend at most $\$65$ in a single month, what is the maximum number of fitness classes the member can attend?

Show answer & explanation

Answer: 6

Answer

The maximum number of fitness classes the member can attend is

6

.

Let

c

represent the number of classes. The inequality representing the budget constraint is

35 + 5c \le 65

. Subtracting

35

from both sides of the inequality yields

5c \le 30

. Dividing both sides by

5

yields

c \le 6

. Since the number of classes must be a whole number, the maximum number of classes the member can attend is

6

.

Step-by-Step Solution

1

Set up the inequality representing the situation.

35 + 5c \le 65

The cost of the membership is a fixed

\$35

plus

\$5

per class

c

, and this total must not exceed

\$65

.

2

Subtract

35

from both sides of the inequality to isolate the variable term.

5c \le 30

Isolating the term with the variable allows us to solve for

c

.

3

Divide both sides by

5

to find the solution range for the number of classes.

c \le 6

Dividing by the coefficient of the variable gives the upper limit for the number of classes.

Key Concept

Solving one-variable linear inequalities in a real-world context.

Question 131Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(k, 12)$ and $(8, k)$ , where $k$ is a constant. If the slope of the graph of $f$ is $-\frac{1}{3}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 14

Answer

14

The slope of a line passing through

(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

(k, 12)

and

(8, k)

and the slope

-\frac{1}{3}

into the formula gives

-\frac{1}{3} = \frac{k - 12}{8 - k}

. Cross-multiplying yields

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

, which simplifies to

3k - 36 = -8 + k

. Isolating the variable

k

gives

2k = 28

, so

k = 14

.

Step-by-Step Solution

1

Apply the slope formula

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

using the points

(k, 12)

and

(8, k)

with slope

-\frac{1}{3}

.

-\frac{1}{3} = \frac{k - 12}{8 - k}

The slope of a linear function is constant and defined by the ratio of the change in

y

-values to the change in

x

-values.

2

Cross-multiply to solve the equation for

k

.

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

To eliminate the fractions and solve the rational equation.

3

Distribute and simplify both sides of the equation.

3k - 36 = -8 + k

Distributing the constants on both sides prepares the equation for isolating the variable

k

.

4

Isolate the variable

k

by subtracting

k

from both sides and adding

36

to both sides.

2k = 28

Grouping like terms on opposite sides of the equation.

5

Divide by

2

to find the value of

k

.

k = 14

To find the final numerical value of the constant

k

.

Key Concept

Linear function slope formula

Estimated Time:1m 30s

Question 132Question →

An artist sells customized prints online. The total price, $y$ , in dollars, for an order of $x$ prints is given by the equation $y = 8.50x + 4.00$ , where $4.00$ represents a flat shipping fee. If a customer's total order price is $38.00$ , how many prints did the customer order?

Show answer & explanation

Answer: 4

Answer

4

Substituting

38.00

for

y

in the given linear equation

y = 8.50x + 4.00

yields

38.00 = 8.50x + 4.00

. Subtracting

4.00

from both sides of the equation results in

34.00 = 8.50x

. Dividing both sides of the equation by

8.50

yields

x = 4

. Therefore, the customer ordered 4 prints.

Step-by-Step Solution

1

Substitute the given total price of

38.00

for

y

in the equation.

38.00 = 8.50x + 4.00

The variable

y

represents the total price of the order in dollars, which is given as

38.00

.

2

Subtract the shipping fee of

4.00

from both sides of the equation.

34.00 = 8.50x

Subtracting

4.00

from both sides isolates the variable term

8.50x

.

3

Divide both sides of the equation by

8.50

.

x = 4

Dividing the remaining cost of

34.00

by the unit price per print of

8.50

yields the total number of prints ordered.

Key Concept

Solving a linear equation in two variables for one variable when the value of the other variable is given.

Estimated Time:45s

Question 133Question →

In the system of equations below, $a$ and $b$ are constants.

\begin{aligned} 3x - 4y &= 7 \\ ax - 12y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of $a + b$ ?

Show answer & explanation

Answer: 30

Answer

30

For a system of two linear equations in two variables to have infinitely many solutions, the two equations must represent the same line. Comparing the

y

-coefficients shows that

-12

is

3

times

-4

. Multiplying the first equation by

3

gives the equivalent equation

9x - 12y = 21

. For this to be identical to the second equation,

ax - 12y = b

, the corresponding coefficients must be equal, meaning

a = 9

and

b = 21

. Therefore, the value of

a + b

is

9 + 21 = 30

.

Step-by-Step Solution

1

Identify the condition for a system of linear equations to have infinitely many solutions.

The two equations must represent the same line, making them equivalent equations.

Infinitely many solutions occur when the two equations in a system are scalar multiples of each other.

2

Align the coefficients of the

y

-terms in both equations.

Multiplying the first equation by

3

gives

9x - 12y = 21

.

The coefficient of

y

in the second equation is

-12

, which is

3

times the coefficient of

y

in the first equation (

-4

).

3

Equate the coefficients of the remaining terms to find the values of

a

and

b

.

a = 9

and

b = 21

.

For

9x - 12y = 21

and

ax - 12y = b

to be equivalent, their corresponding coefficients and constants must be equal.

4

Calculate the value of

a + b

.

a + b = 9 + 21 = 30

.

The problem asks for the value of the sum

a + b

.

Key Concept

Determining constants in a system of linear equations with infinitely many solutions

Estimated Time:1m 30s

Question 134Question →

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

$x$	$f(x)$
$1$	$5$
$4$	$17$
$7$	$29$

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

4

\frac{1}{4}

3

5

Show answer & explanation

Answer: $4$

Answer

The slope of the graph of

f

in the

xy

-plane is

4

.

The slope of a linear function can be found using any two points from the table,

(x_1, y_1)

and

(x_2, y_2)

, with the formula

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

. Using the points

(1, 5)

and

(4, 17)

, the slope is

\frac{17 - 5}{4 - 1} = 4

.

Step-by-Step Solution

1

Select two points from the table to calculate the slope.

Using the points

(1, 5)

and

(4, 17)

, we have

x_1 = 1

,

y_1 = 5

,

x_2 = 4

, and

y_2 = 17

.

Any two points on a line can be used to find its constant slope.

2

Apply the slope formula

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

.

m = \frac{17 - 5}{4 - 1}

The slope formula calculates the ratio of the change in the vertical direction (

y

) to the change in the horizontal direction (

x

).

3

Simplify the expression to find the final slope value.

m = \frac{12}{3} = 4

Subtracting the coordinates and dividing gives the simplified slope.

Key Concept

Calculating the slope of a linear function from a table of values.

Estimated Time:45s

Question 135Question →

In the $xy$ -plane, the graphs of two linear functions, $f$ and $g$ , are perpendicular lines that intersect at the point $(12, k)$ , where $k$ is a positive constant. If the $y$ -intercept of the graph of $f$ is $(0, 24)$ and the $y$ -intercept of the graph of $g$ is $(0, -6)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

18

To find the value of

k

, we determine the slopes of the two lines. The line representing

f

passes through

(0, 24)

and

(12, k)

, so its slope is

m_f = \frac{k - 24}{12}

. The line representing

g

passes through

(0, -6)

and

(12, k)

, so its slope is

m_g = \frac{k + 6}{12}

. Since the two lines are perpendicular, the product of their slopes is

-1

. This gives the equation

\left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1

. Multiplying both sides by

144

yields

(k - 24)(k + 6) = -144

. Expanding the left side gives

k^2 - 18k - 144 = -144

. Adding

144

to both sides results in

k^2 - 18k = 0

, which factors as

k(k - 18) = 0

. Since

k

is a positive constant,

k

must be

18

.

Step-by-Step Solution

1

Determine the slopes of the lines

f

and

g

in terms of

k

.

The slope of

f

is

m_f = \frac{k - 24}{12 - 0} = \frac{k - 24}{12}

. The slope of

g

is

m_g = \frac{k - (-6)}{12 - 0} = \frac{k + 6}{12}

.

We use the slope formula

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

for each line passing through their respective

y

-intercepts and their intersection point

(12, k)

.

2

Set up an equation using the perpendicular condition.

m_f \cdot m_g = -1 \implies \left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1 \implies \frac{(k - 24)(k + 6)}{144} = -1

Since the graphs of

f

and

g

are perpendicular, the product of their slopes must equal

-1

.

3

Solve the quadratic equation for

k

.

(k - 24)(k + 6) = -144 \implies k^2 - 18k - 144 = -144 \implies k^2 - 18k = 0 \implies k(k - 18) = 0

. Since

k

must be positive,

k = 18

.

Multiplying by

144

, expanding, and factoring the quadratic equation gives the possible values

k = 0

and

k = 18

. The problem states

k

is positive, so

k = 18

.

Key Concept

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

-1

.

Question 136Question →

In the $xy$ -plane, a line with a negative slope passes through the point $(4, 3)$ and intersects the positive $x$ -axis at $(a, 0)$ and the positive $y$ -axis at $(0, b)$ . If the area of the triangle formed by this line and the coordinate axes is $24$ , what is the value of $b$ ?

Show answer & explanation

Answer: 6

Answer

The value of

b

is 6.

The correct value is 6. By representing the linear equation in intercept form as

\frac{x}{a} + \frac{y}{b} = 1

and substituting the given point

(4, 3)

, we obtain

\frac{4}{a} + \frac{3}{b} = 1

. Using the area of the triangle,

\frac{1}{2}ab = 24

, we can write

a = \frac{48}{b}

. Substituting this into the intercept equation yields

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

leads to the quadratic equation

b^2 - 12b + 36 = 0

, which factors as

(b-6)^2 = 0

, giving the unique solution

b = 6

.

Step-by-Step Solution

1

Express the line in intercept form

\frac{x}{a} + \frac{y}{b} = 1

Since the intercepts are

(a, 0)

and

(0, b)

with

a > 0

and

b > 0

, the intercept form of a linear equation is the most direct representation.

2

Substitute the given point

(4, 3)

into the intercept form

\frac{4}{a} + \frac{3}{b} = 1

The line passes through

(4, 3)

, so these coordinates must satisfy the equation of the line.

3

Express

a

in terms of

b

using the area of the triangle

a = \frac{48}{b}

The area of the right triangle with base

a

and height

b

is

\frac{1}{2}ab = 24

, which gives

ab = 48

.

4

Substitute

a = \frac{48}{b}

into the equation from Step 2

\frac{b}{12} + \frac{3}{b} = 1

This reduces the equation to a single variable

b

.

5

Clear denominators and write the equation in standard quadratic form

b^2 - 12b + 36 = 0

Multiplying both sides by

12b

allows us to form a quadratic equation.

6

Factor the quadratic equation to solve for

b

b = 6

The quadratic expression is a perfect square trinomial,

(b-6)^2 = 0

, which yields

b = 6

.

Key Concept

Using the intercept form of a linear equation and geometric properties of linear graphs to solve for unknown parameters.

Question 137Question →

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

Show answer & explanation

Answer: 11

Answer

11

Evaluating the linear equation by clearing the fractions with a common denominator of 10 gives

6(2x - 4) - 5(x - 3) = 12

. Distributing terms yields

12x - 24 - 5x + 15 = 12

. Combining like terms results in

7x - 9 = 12

. Solving for

x

gives

x = 3

. Finally, substituting

x = 3

into the target expression

5x - 4

yields

5(3) - 4 = 11

.

Step-by-Step Solution

1

Multiply both sides of the equation by 10 to clear the denominators.

6(2x - 4) - 5(x - 3) = 12

The least common denominator of 5 and 2 is 10. Multiplying both sides by 10 simplifies the equation by removing the fractions.

2

Distribute the constants 6 and -5 into the parentheses.

12x - 24 - 5x + 15 = 12

Applying the distributive property removes the parentheses. Be careful to distribute the negative sign for the second term:

-5 \cdot (x - 3) = -5x + 15

.

3

Combine like terms on the left side of the equation.

7x - 9 = 12

Grouping the

x

terms (

12x - 5x = 7x

) and the constant terms (

-24 + 15 = -9

) simplifies the equation.

4

Isolate the variable term

7x

and solve for

x

.

x = 3

Adding 9 to both sides gives

7x = 21

. Dividing both sides by 7 yields

x = 3

.

5

Substitute the value of

x

into the expression

5x - 4

.

11

The question asks for the value of

5x - 4

, not just

x

, so we evaluate

5(3) - 4 = 11

.

Key Concept

Solving linear equations in one variable by clearing fractions, distributing terms, and isolating the variable.

Question 138Question →

A linear model is used to estimate the height of a plant, in centimeters, based on the number of weeks since it was planted. According to the model, the plant's height is $5$ centimeters at week $2$ , and its height is $8$ centimeters at week $8$ . Which of the following is the predicted height of the plant, in centimeters, at week $16$ ?

14

29

12

47

Show answer & explanation

Answer: $12$

Answer

The predicted height of the plant at week 16 is 12 centimeters.

The correct answer is 12. To find the predicted height, we determine the linear model equation. The slope is calculated as

m = \frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

. Substituting the point

(2, 5)

into the slope-intercept form

y = mx + b

gives

5 = 0.5(2) + b

, which simplifies to

b = 4

. The linear model is

y = 0.5x + 4

. Substituting

16

for

x

yields

y = 0.5(16) + 4 = 8 + 4 = 12

.

Step-by-Step Solution

1

Identify two data points from the given information to represent the linear relationship.

The coordinates are

(2, 5)

and

(8, 8)

where the x-coordinate represents the week and the y-coordinate represents the height of the plant in centimeters.

A linear relationship is uniquely determined by two points, which allows us to find the slope and equation of the line.

2

Calculate the slope of the line, which represents the constant growth rate of the plant.

The slope

m

is

\frac{8 - 5}{8 - 2} = \frac{3}{6} = 0.5

centimeters per week.

The slope represents the constant rate of change of the plant's height per week.

3

Determine the y-intercept of the linear model using one of the coordinates.

Using the point

(2, 5)

and the equation

y = mx + b

, we get

5 = 0.5(2) + b

, which simplifies to

b = 4

.

Finding the y-intercept allows us to write the complete equation for the linear function representing the plant's growth.

4

Substitute the target week into the linear function to predict the height of the plant.

For week 16,

y = 0.5(16) + 4 = 8 + 4 = 12

centimeters.

Evaluating the function at

x = 16

yields the predicted height of the plant at that specific time.

Key Concept

Finding and evaluating linear functions using two given coordinate points.

Estimated Time:1m 30s

Question 139Question →

For a linear function $f$ , the equation $f(x + 3) - f(x - 1) = 16$ is true for all real numbers $x$ . If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 5)$ , what is the $x$ -intercept of the graph of $f$ ?

-\frac{3}{4}

\frac{3}{4}

\frac{11}{8}

-18

Show answer & explanation

Answer: $\frac{3}{4}$

Answer

The

x

-intercept of the graph of

f

is

\frac{3}{4}

.

By writing the linear function as

f(x) = mx + b

, we can evaluate

f(x+3) - f(x-1)

as

(mx + 3m + b) - (mx - m + b) = 4m

. Setting this equal to the given value of

16

yields a slope of

m = 4

. Using the point

(2, 5)

in

f(2) = 4(2) + b = 5

gives the

y

-intercept

b = -3

, so the function is

f(x) = 4x - 3

. The

x

-intercept is found by setting

f(x) = 0

, which yields

x = \frac{3}{4}

.

Step-by-Step Solution

1

Represent the linear function in slope-intercept form

f(x) = mx + b

and substitute the expressions

x+3

and

x-1

into the function.

f(x+3) = m(x+3) + b

and

f(x-1) = m(x-1) + b

.

This allows us to evaluate the given functional equation using the general parameters of a linear function.

2

Calculate the difference

f(x+3) - f(x-1)

and set it equal to

16

to find the slope

m

.

(mx + 3m + b) - (mx - m + b) = 4m = 16

, which simplifies to

m = 4

.

The difference between two values of a linear function is directly proportional to the difference in their inputs, where the constant of proportionality is the slope.

3

Substitute the point

(2, 5)

and the slope

m = 4

into

f(x) = 4x + b

to find the

y

-intercept

b

.

5 = 4(2) + b \implies 5 = 8 + b \implies b = -3

.

A linear function is completely defined once both its slope and a point on its graph are known.

4

Find the

x

-intercept of the graph by setting

f(x) = 0

and solving for

x

.

4x - 3 = 0 \implies x = \frac{3}{4}

.

The

x

-intercept of a function's graph is the input value at which the function's output equals zero.

Key Concept

Determining the slope, equation, and intercepts of a linear function from functional relationships and coordinate points.

Estimated Time:2m 0s

Question 140Question →

\begin{cases} y \geq -2x + 4 \\ y < \frac{1}{2}x + 2 \end{cases}

Which of the following ordered pairs $(x, y)$ is a solution to the system of inequalities above?

(1, 2)

(0, -1)

(2, 3)

(0, 3)

Show answer & explanation

Answer: $(1, 2)$

Answer

The ordered pair

(1, 2)

is a solution to the system of inequalities.

Substituting the coordinates of the point stating

(1, 2)

into the system of inequalities produces two true statements:

2 \geq -2(1) + 4

simplifies to

2 \geq 2

, which is true; and

2 < \frac{1}{2}(1) + 2

simplifies to

2 < 2.5

, which is also true. Therefore, the ordered pair

(1, 2)

is a solution to the system.

Step-by-Step Solution

1

Substitute the coordinates of each option into the first inequality,

y \geq -2x + 4

.

For

(1, 2)

,

2 \geq 2

(True). For

(0, -1)

,

-1 \geq 4

(False). For

(2, 3)

,

3 \geq 0

(True). For

(0, 3)

,

3 \geq 4

(False).

To determine which ordered pairs satisfy the first inequality.

2

Substitute the coordinates of the remaining candidate pairs,

(1, 2)

and

(2, 3)

, into the second inequality,

y < \frac{1}{2}x + 2

.

For

(1, 2)

,

2 < 2.5

(True). For

(2, 3)

,

3 < 3

(False).

To determine which of the remaining candidates also satisfies the second inequality.

3

Identify the ordered pair that satisfies both inequalities in the system.

Only

(1, 2)

satisfies both inequalities.

An ordered pair is a solution to a system of inequalities if and only if it satisfies all inequalities in the system.

Key Concept

Verifying solutions to a system of linear inequalities by substitution.

Algebra

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