Algebra Practice Questions — SAT — Page 6

If the equations $x + 2y = 12$ and $x + y = 7$ are true for the same values of $x$ and $y$ , what is the value of $y$ ?

Show answer & explanation

Answer: 5

Answer

The value of

y

is 5.

Subtracting the equation

x + y = 7

from

x + 2y = 12

gives

(x - x) + (2y - y) = 12 - 7

, which simplifies directly to

y = 5

.

Step-by-Step Solution

1

Subtract the second equation from the first equation.

y = 5

Subtracting

x + y = 7

from

x + 2y = 12

eliminates the

x

terms, leaving a single linear equation in terms of

y

.

Key Concept

Solving systems of linear equations using elimination.

Question 102Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(6, 4)$ . Line $g$ is perpendicular to the graph of $f$ and has a $y$ -intercept of $(0, 12)$ . The graph of $f$ and line $g$ intersect at the point $P(h, k)$ , where $h$ and $k$ are integers. If the slope of the graph of $f$ is a positive integer greater than $5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 11

Answer

The value of

k

is

11

.

By equating the linear equations representing

f

and

g

, we find that the intersection point's

x

-coordinate

h

is given by

h = 6 + \frac{8m - 6}{m^2 + 1}

. Since

h

is an integer and

m

is a positive integer greater than

5

, the inequality

m^2 + 1 \le 8m - 6

limits

m

to the range

1 \le m \le 7

. Testing the integers greater than

5

(which are

6

and

7

) reveals that only

m = 7

yields an integer value for

h

, resulting in

h = 7

. Substituting these values back into the equation of either line yields

k = 11

.

Step-by-Step Solution

1

Write the equation of the linear function

f

in terms of its slope

m

.

y - 4 = m(x - 6) \implies y = mx + 4 - 6m

Since the graph of

f

passes through the point

(6, 4)

with slope

m

, we can use the point-slope form of a linear equation.

2

Write the equation of line

g

in slope-intercept form.

y = -\frac{1}{m}x + 12

Line

g

is perpendicular to the graph of

f

, so its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Its

y

-intercept is

(0, 12)

.

3

Equate the two equations to find the

x

-coordinate

h

of the intersection point

P(h, k)

in terms of

m

.

h = \frac{6m^2 + 8m}{m^2 + 1}

At the intersection point, the

y

-values of both equations are equal, so we solve

mx + 4 - 6m = -\frac{1}{m}x + 12

for

x

.

4

Analyze the expression for

h

to find the integer constraints on

m

.

h = 6 + \frac{8m - 6}{m^2 + 1} \implies m^2 - 8m + 7 \le 0 \implies 1 \le m \le 7

For

h

to be an integer,

\frac{8m - 6}{m^2 + 1}

must be an integer. Since

m

is a positive integer greater than

5

, the numerator is positive, meaning the denominator

m^2 + 1

must be less than or equal to the numerator

8m - 6

.

5

Test the possible integer values of

m

greater than

5

to find the one that yields integer coordinates.

m = 7 \implies h = 7

The only integers greater than

5

in the range

[1, 7]

are

6

and

7

. Testing

m=6

gives a non-integer, while

m=7

gives

h = 7

, which is an integer.

6

Calculate the

y

-coordinate

k

of the intersection point

P

.

k = 11

Substitute

h = 7

into the equation of line

g

:

k = -\frac{1}{7}(7) + 12 = 11

.

Key Concept

Linear Functions and Graphs

Question 103Question →

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $m$ and a $y$ -intercept of $(0, b)$ . The graph of the linear function $g$ has a slope of $-3m$ and a $y$ -intercept of $(0, b + 15)$ . If the graphs of $f$ and $g$ intersect at the point $(3, 5)$ , what is the value of $f(5)$ ?

Show answer & explanation

Answer: 7.5

Answer

7.5 (or 15/2)

By writing the equations of the two lines as

f(x) = mx + b

and

g(x) = -3mx + b + 15

, and substituting the intersection point

(3, 5)

, we get the system of equations

3m + b = 5

and

-9m + b = -10

. Subtracting these equations gives

12m = 15

, which simplifies to

m = 1.25

. Substituting

m

back into

3m + b = 5

yields

b = 1.25

. The function is therefore

f(x) = 1.25x + 1.25

. Evaluating this at

x = 5

gives

f(5) = 1.25(5) + 1.25 = 7.5

.

Step-by-Step Solution

1

Formulate equations for both linear functions using slope-intercept form.

f(x) = mx + b

and

g(x) = -3mx + b + 15

The equation of a line with slope

M

and

y

-intercept

(0, B)

is given by

y = Mx + B

.

2

Substitute the coordinates of the intersection point

(3, 5)

into both equations.

3m + b = 5

and

-9m + b = -10

Since the intersection point lies on both lines, its coordinates must satisfy both equations.

3

Solve the system of equations for

m

and

b

.

m = 1.25

and

b = 1.25

Subtracting the equations eliminates

b

, giving

12m = 15 \implies m = 1.25

. Substituting this back into the first equation gives

3(1.25) + b = 5 \implies b = 1.25

.

4

Evaluate

f(5)

using the found values.

f(5) = 1.25(5) + 1.25 = 7.5

Substitute

x = 5

into the equation

f(x) = 1.25x + 1.25

.

Key Concept

Solving systems of linear equations and evaluating linear functions from graph intersections.

Estimated Time:2m 0s

Question 104Question →

In the $xy$ -plane, a line passes through the origin and has a slope of $\frac{3}{5}$ . If the point $(k, 9)$ lies on the line, what is the value of $k$ ?

Show answer & explanation

Answer: 15

Answer

15

A line passing through the origin has a

y

-intercept of

0

. Thus, its equation in slope-intercept form is

y = mx

, where

m

is the slope. Given the slope is

\frac{3}{5}

, the equation is

y = \frac{3}{5}x

. Since the point

(k, 9)

lies on the line, substituting these coordinates yields

9 = \frac{3}{5}k

. Multiplying both sides by

\frac{5}{3}

gives

k = 15

.

Step-by-Step Solution

1

Write the equation of the line in slope-intercept form.

y = \frac{3}{5}x

The line passes through the origin

(0,0)

, so the

y

-intercept is

0

, and the slope is

\frac{3}{5}

.

2

Substitute the coordinates of the point

(k, 9)

into the line's equation.

9 = \frac{3}{5}k

A point lies on a line if its coordinates satisfy the equation of the line.

3

Solve the equation for

k

.

k = 15

Multiply both sides of the equation by

\frac{5}{3}

to isolate

k

.

Key Concept

Using the slope and a point on a line to determine an unknown coordinate.

Question 105Question →

A system of two linear equations is given below.

y = x - 2

2x - 3y = 1

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

5

-7

-5

7

Show answer & explanation

Answer: 5

Answer

The value of

x

is 5.

To find the value of

x

, substitute the expression for

y

from the first equation into the second equation:

2x - 3(x - 2) = 1

. Distribute the

-3

to obtain

2x - 3x + 6 = 1

, which simplifies to

-x + 6 = 1

. Subtracting 6 from both sides gives

-x = -5

. Multiplying or dividing both sides by

-1

yields

x = 5

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

2x - 3(x - 2) = 1

Substituting the expression reduces the system of two equations to one linear equation containing only the variable

x

.

2

Distribute the coefficient

-3

to both terms inside the parentheses.

2x - 3x + 6 = 1

Applying the distributive property removes the parentheses so that like terms can be combined.

3

Combine like terms and isolate

x

.

-x + 6 = 1 \implies -x = -5 \implies x = 5

Combining the

x

terms and subtracting 6 from both sides isolates

-x

, allowing you to find the final value of

x

.

Key Concept

Solving systems of linear equations using the substitution method

Alternative Method

Alternatively, you can solve this system using elimination. First, rewrite the first equation in standard form by subtracting

x

from both sides, which gives

-x + y = -2

. Next, multiply this equation by 3 to align coefficients:

-3x + 3y = -6

. Adding this result directly to the second equation,

2x - 3y = 1

, eliminates the

y

terms and leaves

-x = -5

, which simplifies to

x = 5

.

Estimated Time:45s

Question 106Question →

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

4(x - 2) - a(3 - 2x) = 8x - 5

If the equation has no solution for $x$ , what is the value of $a$ ?

-2

2

4

6

Show answer & explanation

Answer: 2

Answer

2

For the linear equation to have no solution, the coefficients of the

x

terms on both sides of the equation must be equal, while the constant terms must be different. First, expand the left side of the equation:

4(x - 2) - a(3 - 2x) = 4x - 8 - 3a + 2ax

. Grouping the

x

terms and constant terms gives

(4 + 2a)x - (8 + 3a) = 8x - 5

. Setting the coefficient of

x

on the left side equal to the coefficient of

x

on the right side gives

4 + 2a = 8

. Solving for

a

yields

2a = 4

, which simplifies to

a = 2

. Substituting

a = 2

back into the equation yields

8x - 14 = 8x - 5

, or

-14 = -5

, which is a false statement with no solution. Therefore, the value of

a

is

2

.

Step-by-Step Solution

1

Expand and simplify the left side of the equation.

The left side expands to

4x - 8 - 3a + 2ax

. Grouping the terms by the variable

x

gives

(4 + 2a)x - (8 + 3a)

.

To find when the linear equation has no solution, we need to rewrite it in the standard form

px + q = rx + s

by distributing terms and grouping like terms.

2

Equate the coefficients of the

x

terms from both sides of the equation.

4 + 2a = 8

A linear equation of the form

px + q = rx + s

has no solution if the coefficients of the variable are equal (

p = r

) but the constants are not (

q \neq s

).

3

Solve for the constant

a

.

2a = 4 \implies a = 2

Subtracting

4

from both sides of

4 + 2a = 8

isolates the variable term, and dividing by

2

gives the value of

a

.

4

Verify that the constants are different when

a = 2

.

Substituting

a = 2

back into the constant terms yields a left-side constant of

-(8 + 3(2)) = -14

and a right-side constant of

-5

. Since

-14 \neq -5

, the equation has no solution.

If the constant terms were also equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Linear Equations in One Variable (No Solution Case)

Question 107Question →

A system of equations is shown below.

y = -2(x - 5)

3x + y = 13

What is the value of $x$ ?

23

4

3

8

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

Substituting the expression for

y

from the first equation into the second equation gives

3x - 2(x - 5) = 13

. Distributing the

-2

yields

3x - 2x + 10 = 13

. Combining the

x

terms gives

x + 10 = 13

. Subtracting 10 from both sides of the equation yields

x = 3

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

3x - 2(x - 5) = 13

This substitution reduces the system to a single linear equation in terms of

x

.

2

Apply the distributive property to simplify the equation.

3x - 2x + 10 = 13

Multiplying

-2

by

x

gives

-2x

, and multiplying

-2

by

-5

gives

+10

.

3

Combine like terms and isolate

x

.

x + 10 = 13

, which simplifies to

x = 3

Subtracting 10 from both sides gives the value of

x

.

Key Concept

Solving a system of linear equations using substitution.

Estimated Time:1m 0s

Question 108Question →

A line in the coordinate plane is defined by the function $f(x) = px + q$ , where $p$ and $q$ are constants. This line contains the point $(6, 2)$ and intersects the vertical axis at a point with a negative $y$ -coordinate. If the region bounded by the line and the two coordinate axes is a right triangle with an area of $25$ , what is the value of $p$ ?

\frac{1}{18}

18

2

50

Show answer & explanation

Answer: 2

Answer

2

The correct answer is the option representing the value

2

. Substituting the point

(6, 2)

into the linear function gives

q = 2 - 6p

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

\frac{1}{2} \times |\text{base}| \times |\text{height}| = \frac{q^2}{2p}

. Setting this equal to

25

leads to the quadratic equation

18p^2 - 37p + 2 = 0

. Factoring this equation yields

p = 2

or

p = \frac{1}{18}

. Checking the

y

-intercept condition (

q < 0

) shows that only

p = 2

yields a negative

y

-intercept (

q = -10

), whereas

p = \frac{1}{18}

yields a positive

y

-intercept (

q = \frac{5}{3}

).

Step-by-Step Solution

1

Use the point

(6, 2)

to express

q

in terms of

p

.

q = 2 - 6p

Since the line contains

(6, 2)

, substituting

x = 6

and

f(x) = 2

into the function

f(x) = px + q

must satisfy the equation.

2

Determine the intercepts of the line on the coordinate axes.

The vertical intercept is

(0, q)

and the horizontal intercept is

\left(-\frac{q}{p}, 0\right)

.

The vertical intercept is found by setting

x = 0

, and the horizontal intercept is found by setting

f(x) = 0

.

3

Express the area of the bounded right triangle and set it equal to

25

.

\frac{q^2}{2p} = 25 \Rightarrow q^2 = 50p

The area of a right triangle with vertices at the origin and the intercepts is

\frac{1}{2} \times |\text{base}| \times |\text{height}| = \frac{1}{2} \left(-\frac{q}{p}\right)(-q) = \frac{q^2}{2p}

.

4

Substitute

q = 2 - 6p

into

q^2 = 50p

and solve the quadratic equation.

18p^2 - 37p + 2 = 0 \Rightarrow (18p - 1)(p - 2) = 0 \Rightarrow p = 2

or

p = \frac{1}{18}

.

Substituting the expression for

q

gives a quadratic equation in terms of

p

that can be factored to find its roots.

5

Verify which root satisfies the condition that the vertical intersection point has a negative

y

-coordinate.

For

p = 2

,

q = -10

(valid). For

p = \frac{1}{18}

,

q = \frac{5}{3}

(invalid). Thus,

p = 2

.

The problem states that the vertical axis intersection (

y

-intercept) must have a negative

y

-coordinate (

q < 0

).

Key Concept

Using coordinate points and geometric properties of intercepts to determine parameters of linear functions

Question 109Question →

In the $xy$ -plane, the graph of a linear function $f$ has a negative slope and a positive $y$ -intercept. The graph of $f$ intersects the $x$ -axis at point $A$ and the $y$ -axis at point $B$ . A second line, $L$ , is perpendicular to the graph of $f$ and passes through the origin. If line $L$ intersects the graph of $f$ at point $P$ such that the ratio of the area of triangle $OAP$ to the area of triangle $OBP$ is $4:9$ , what is the slope of the graph of $f$ ?

-\frac{3}{2}

-\frac{2}{3}

-\frac{9}{4}

-\frac{4}{9}

Show answer & explanation

Answer: $-\frac{3}{2}$

Answer

The slope of the graph of

f

is

-\frac{3}{2}

.

The correct answer is

-\frac{3}{2}

. By setting up the equations of the lines

f(x) = mx + b

and

y = -\frac{1}{m}x

, we can express the coordinates of the intercepts

A

and

B

, as well as the intersection point

P

, in terms of

m

and

b

. Calculating the areas of triangle

OAP

and triangle

OBP

gives

\text{Area}(OAP) = \frac{1}{2} \left(-\frac{b}{m}\right) y_0

and

\text{Area}(OBP) = \frac{1}{2} b x_0

. Since

P

lies on the perpendicular line,

y_0 = -\frac{1}{m}x_0

. Substituting this into the area ratio yields

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = \frac{1}{m^2}

. Equating this to the given ratio

\frac{4}{9}

results in

m^2 = \frac{9}{4}

. Given that the slope is negative,

m

must be

-\frac{3}{2}

.

Step-by-Step Solution

1

Define the linear function

f(x)

and the line

L

perpendicular to it.

Let the function be

f(x) = mx + b

, where

m < 0

and

b > 0

. The line

L

perpendicular to the graph of

f

that passes through the origin has the equation

y = -\frac{1}{m}x

.

This establishes the equations of both lines in terms of the slope

m

and

y

-intercept

b

.

2

Determine the coordinates of the intercepts

A

and

B

, and the intersection point

P(x_0, y_0)

.

The

x

-intercept is

A\left(-\frac{b}{m}, 0\right)

and the

y

-intercept is

B(0, b)

. The intersection point

P(x_0, y_0)

is the solution to

mx + b = -\frac{1}{m}x

, which gives

x_0 = -\frac{mb}{m^2+1}

and

y_0 = \frac{b}{m^2+1}

.

Finding these points allows us to express the dimensions of triangles

OAP

and

OBP

.

3

Calculate the areas of triangles

OAP

and

OBP

and find their ratio in terms of

m

.

The base of triangle

OAP

along the

x

-axis is

OA = -\frac{b}{m}

and its height is

y_0

. Thus,

\text{Area}(OAP) = \frac{1}{2} \left(-\frac{b}{m}\right) y_0

. The base of triangle

OBP

along the

y

-axis is

OB = b

and its height is

x_0

. Thus,

\text{Area}(OBP) = \frac{1}{2} b x_0

. The ratio of their areas is

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = \frac{-\frac{b}{m} y_0}{b x_0} = -\frac{y_0}{m x_0}

. Since

P

lies on

L

, we have

\frac{y_0}{x_0} = -\frac{1}{m}

. Substituting this gives the ratio

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = -\frac{1}{m}\left(-\frac{1}{m}\right) = \frac{1}{m^2}

.

This simplifies the geometric relationship to a direct relation between the area ratio and the slope of the function.

4

Solve for the slope

m

using the given ratio of

4:9

.

Setting

\frac{1}{m^2} = \frac{4}{9}

yields

m^2 = \frac{9}{4}

. Since the problem states the slope is negative, we take the negative square root to get

m = -\frac{3}{2}

.

This identifies the correct slope value matching the given constraints.

Key Concept

Using coordinate geometry and system of linear equations to determine slopes and intersections, and relating those to geometric areas on the coordinate plane.

Question 110Question →

A project manager uses the linear equation $ax + by = c$ to model the allocation of resources, where $x$ represents the number of hours spent on research, $y$ represents the number of hours spent on development, and $a, b,$ and $c$ are positive constants. Initially, the project manager allocates $40$ hours for research and $30$ hours for development. To optimize the project, the manager decides to increase the research hours by $25\%$ and decrease the development hours by $20\%$ such that the total resource value, $c$ , remains unchanged. Which of the following equations correctly expresses $a$ in terms of $b$ ?

a = 1.67b

a = 0.6b

a = 0.64b

a = 1.6b

Show answer & explanation

Answer: $a = 0.6b$

Answer

The equation

a = 0.6b

correctly expresses

a

in terms of

b

.

The correct equation is

a = 0.6b

. The initial resource allocation is represented by

40a + 30b = c

. Increasing

40

by

25\%

gives

50

, and decreasing

30

by

20\%

gives

24

, which yields the new equation

50a + 24b = c

. Equating these two expressions since

c

is constant gives

40a + 30b = 50a + 24b

. Simplifying this equation results in

6b = 10a

, which gives

a = 0.6b

.

Step-by-Step Solution

1

Write the linear equation representing the initial allocation of resources.

40a + 30b = c

The initial values are

40

hours for research (

x

) and

30

hours for development (

y

).

2

Calculate the new values of

x

and

y

after the percentage changes are applied.

x = 40 \times 1.25 = 50

and

y = 30 \times 0.80 = 24

Research hours increase by

25\%

and development hours decrease by

20\%

.

3

Write the new linear equation with the updated resource allocations.

50a + 24b = c

The total resource value

c

remains unchanged.

4

Equate the two expressions for

c

and solve for

a

in terms of

b

.

40a + 30b = 50a + 24b \implies 6b = 10a \implies a = 0.6b

Since both equations equal the same constant

c

, we set them equal to each other and isolate the variable

a

.

Key Concept

Expressing and manipulating linear relationships in two variables with constraints.

Alternative Method

Instead of keeping

c

general, we can choose a convenient value for

c

, such as

c = 120

. This gives two linear equations in terms of

a

and

b

:

40a + 30b = 120

and

50a + 24b = 120

. Solving this system for the ratio of

a

to

b

yields

a = 1.33

and

b = 2.22

, which confirms that

a/b = 0.6

, or

a = 0.6b

.

Estimated Time:3m 0s

Question 111Question →

A student tracks the remaining battery percentage, $y$ , of a tablet after $x$ hours of use. The table below shows the battery percentage at three different times:

Time ( $x$ , hours)	Battery percentage ( $y$ )
0	90
3	75
6	60

Which of the following equations represents the relationship between $x$ and $y$ ?

y = -15x + 90

y = -\frac{1}{5}x + 90

y = -5x + 90

y = 90x - 5

Show answer & explanation

Answer: $y = -5x + 90$

Answer

y = -5x + 90

The correct equation is

y = -5x + 90

. According to the table, when time

x = 0

, the battery percentage

y = 90

, which indicates that the y-intercept of the line is

90

. The slope can be determined using any two points from the table, such as

(0, 90)

and

(3, 75)

:

\text{slope} = \frac{75 - 90}{3 - 0} = \frac{-15}{3} = -5

. Writing this in slope-intercept form gives

y = -5x + 90

.

Step-by-Step Solution

1

Identify the y-intercept from the table.

The y-intercept

b

is

90

.

When the time

x = 0

hours, the battery percentage

y = 90

. This corresponds to the y-intercept of the linear equation.

2

Calculate the slope (rate of change) using two points from the table.

The slope

m

is

-5

.

Using the points

(0, 90)

and

(3, 75)

, the slope is calculated as the change in

y

divided by the change in

x

:

m = \frac{75 - 90}{3 - 0} = \frac{-15}{3} = -5

.

3

Substitute the slope and y-intercept into the slope-intercept form equation.

y = -5x + 90

Plugging

m = -5

and

b = 90

into

y = mx + b

yields the final equation.

Key Concept

Determining a linear equation in two variables from a table of values.

Estimated Time:1m 0s

Question 112Question →

If the equation $\frac{3}{4}(8x - 12) - \frac{5}{6}(6x - 18) = -\frac{1}{2}(4x - 8)$ is true, what is the value of $x$ ?

\frac{28}{3}

-\frac{10}{3}

-\frac{2}{3}

\frac{2}{3}

Show answer & explanation

Answer: $-\frac{2}{3}$

Answer

-\frac{2}{3}

The correct answer is found by distributing the coefficients across each set of parentheses, combining like terms on the left side to get

x + 6 = -2x + 4

, adding

2x

to both sides to get

3x + 6 = 4

, subtracting

6

from both sides to get

3x = -2

, and finally dividing by

3

to obtain

-\frac{2}{3}

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses on both sides of the equation.

6x - 9 - 5x + 15 = -2x + 4

To eliminate the parentheses and prepare the equation for combining like terms.

2

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

x + 6 = -2x + 4

To simplify the left side of the equation.

3

Add

2x

to both sides of the equation.

3x + 6 = 4

To group all terms containing the variable

x

on the left side.

4

Subtract

6

from both sides, then divide by

3

to solve for

x

.

3x = -2

, which gives

x = -\frac{2}{3}

To isolate

x

and find its value.

Key Concept

Linear Equations in One Variable

Question 113Question →

For the linear function $f$ , the value of $f(0)$ is $3$ and the value of $f(5)$ is $18$ . What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

Show answer & explanation

Answer: 3

Answer

The slope of the graph of

y = f(x)

is

3

.

The slope of a linear function can be determined using any two points

(x_1, y_1)

and

(x_2, y_2)

on its graph with the formula

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

. The given function values

f(0) = 3

and

f(5) = 18

correspond to the coordinate points

(0, 3)

and

(5, 18)

respectively. Substituting these coordinates into the formula gives

m = \frac{18 - 3}{5 - 0} = \frac{15}{5} = 3

.

Step-by-Step Solution

1

Identify the coordinates of two points on the line using the given function values.

The function values

f(0) = 3

and

f(5) = 18

correspond to the points

(0, 3)

and

(5, 18)

on the graph of the function.

In function notation,

f(x) = y

represents a point

(x, y)

on the graph of the function.

2

Calculate the slope using the slope formula.

The slope

m

is calculated as

\frac{18 - 3}{5 - 0} = 3

.

The slope of a line is defined as the change in

y

divided by the change in

x

between any two points on the line.

Key Concept

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

Question 114Question →

If $3(2.5x - 4) - 1.5(x - 2) = 0.8(5x + 10)$ , what is the value of $x$ ?

Show answer & explanation

Answer: 8.5

Answer

8.5

By distributing and simplifying terms, the linear equation simplifies to

2x = 17

, which yields

x = 8.5

.

Step-by-Step Solution

1

Distribute the constants on both sides of the equation.

7.5x - 12 - 1.5x + 3 = 4x + 8

To simplify the expressions.

2

Combine like terms on the left side.

6x - 9 = 4x + 8

To simplify the left side of the equation.

3

Subtract

4x

from both sides of the equation.

2x - 9 = 8

To group variable terms on one side.

4

Add 9 to both sides of the equation.

2x = 17

To isolate the term with the variable.

5

Divide both sides by 2.

x = 8.5

To find the final value of the variable.

Key Concept

Linear Equations in One Variable

Alternative Method

Multiply the entire equation by 10 to clear all decimal values:

30(2.5x - 4) - 15(x - 2) = 8(5x + 10)

. Expanding this yields

75x - 120 - 15x + 30 = 40x + 80

, which simplifies to

60x - 90 = 40x + 80

. Subtracting

40x

and adding 90 to both sides gives

20x = 170

, resulting in

x = 8.5

.

Estimated Time:1m 30s

Question 115Question →

A linear function $f$ is defined by $f(x) = mx + b$ , where $m$ and $b$ are constants. The graph of $f$ in the $xy$ -plane passes through the point $(2, 5)$ . A second linear function $g$ is defined by $g(x) = f(x - 3) + 4$ . The graph of $g$ has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, -k)$ , where $k$ is a nonzero constant. What is the value of $f(8)$ ?

Show answer & explanation

Answer: 11

Answer

11

The correct answer is 11. By using the given point on the graph of

f(x)

, we express its y-intercept

b

in terms of its slope

m

as

b = 5 - 2m

. Substituting this expression into the translation equation

g(x) = f(x - 3) + 4

yields

g(x) = mx - 5m + 9

. Setting the y-intercept of

g(x)

, which is

-5m + 9

, equal to

-k

gives

k = 5m - 9

. Evaluating

g(x)

at its x-intercept

x = k

gives

mk - 5m + 9 = 0

. Substituting the relation

-5m + 9 = -k

into this equation yields

mk - k = 0

, or

k(m - 1) = 0

. Since

k

is nonzero, we find

m = 1

, which in turn gives

b = 3

. Thus, the linear function is

f(x) = x + 3

, and

f(8) = 11

.

Step-by-Step Solution

1

Express the y-intercept constant

b

in terms of the slope

m

.

b = 5 - 2m

Since the graph of

f(x) = mx + b

passes through

(2, 5)

, substituting

x = 2

and

f(x) = 5

gives

5 = 2m + b

.

2

Write the expression for

g(x)

in terms of

m

.

g(x) = mx - 5m + 9

By definition,

g(x) = f(x - 3) + 4 = m(x - 3) + b + 4

. Substituting

b = 5 - 2m

simplifies the expression to

g(x) = mx - 5m + 9

.

3

Relate the y-intercept of

g(x)

to the parameter

k

.

k = 5m - 9

The y-intercept of

g(x)

is

g(0) = -5m + 9

. Since the y-intercept is given as

(0, -k)

, we set

-k = -5m + 9

, which gives

k = 5m - 9

.

4

Set up an equation using the x-intercept of

g(x)

.

g(k) = mk - 5m + 9 = 0

Since the x-intercept of

g(x)

is

(k, 0)

, substituting

x = k

into the expression for

g(x)

must yield

0

.

5

Solve for the slope

m

.

m = 1

Substituting

-5m + 9 = -k

into the equation

mk - 5m + 9 = 0

gives

mk - k = 0

, which factors as

k(m - 1) = 0

. Since

k

is a nonzero constant, we divide by

k

to get

m - 1 = 0

, so

m = 1

.

6

Find the constant

b

and write the final formula for

f(x)

.

f(x) = x + 3

Substituting

m = 1

back into

b = 5 - 2m

yields

b = 3

. Therefore,

f(x) = x + 3

.

7

Evaluate

f(8)

.

11

Substituting

x = 8

into the function

f(x) = x + 3

gives

f(8) = 8 + 3 = 11

.

Key Concept

Linear function transformations and intercept properties

Question 116Question →

The equation $3x + 4y = 36$ represents the relationship between the number of small boxes, $x$ , and large boxes, $y$ , that can fit in a delivery van. If the van is loaded with exactly 6 large boxes, what is the number of small boxes that can also fit in the van?

Show answer & explanation

Answer: 4

Answer

The number of small boxes that can also fit in the van is 4.

Substituting

y = 6

into the equation

3x + 4y = 36

gives

3x + 24 = 36

. Subtracting 24 from both sides results in

3x = 12

. Dividing both sides by 3 yields

x = 4

. Thus, the number of small boxes that can fit is 4.

Step-by-Step Solution

1

Substitute 6 for

y

in the equation

3x + 4y = 36

.

3x + 4(6) = 36

The variable

y

represents the number of large boxes, which is given as 6.

2

Simplify the multiplication in the equation.

3x + 24 = 36

To simplify the terms before solving for

x

.

3

Subtract 24 from both sides of the equation.

3x = 12

To isolate the variable term

3x

on one side of the equation.

4

Divide both sides of the equation by 3.

x = 4

To solve for

x

, the number of small boxes.

Key Concept

Evaluating and solving a linear equation in two variables given the value of one variable.

Estimated Time:45s

Question 117Question →

The graph of the linear function $f$ in the $xy$ -plane is defined by $f(x) = kx + 4$ , where $k$ is a positive constant. The graph of the linear function $g$ is obtained by translating the graph of $f$ left by $3$ units and down by $2$ units. If the $x$ -intercept of the graph of $g$ is $5$ times the $x$ -intercept of the graph of $f$ , what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. The linear function

g(x)

represents the translated graph, which is expressed as

g(x) = f(x+3) - 2

. Substituting

f(x) = kx + 4

yields

g(x) = k(x+3) + 4 - 2 = kx + 3k + 2

. The

x

-intercept of a graph is the point where the function value is

0

. Solving

f(x) = 0

gives the

x

-intercept of the graph of

f

as

x = -\frac{4}{k}

. Solving

g(x) = 0

gives the

x

-intercept of the graph of

g

as

x = -\frac{3k + 2}{k}

. We are given that the

x

-intercept of the graph of

g

is

5

times the

x

-intercept of the graph of

f

, which translates to the equation

-\frac{3k+2}{k} = 5 \left(-\frac{4}{k}\right)

. Multiplying both sides by

-k

(since

k > 0

,

k \neq 0

) yields

3k + 2 = 20

. Subtracting

2

from both sides gives

3k = 18

, and dividing by

3

results in

k = 6

.

Step-by-Step Solution

1

Determine the equation of the translated function

g(x)

g(x) = kx + 3k + 2

Shifting the graph of

f(x)

left by

3

units and down by

2

units translates to

g(x) = f(x+3) - 2

.

2

Find the

x

-intercept of the graph of

f

x = -\frac{4}{k}

The

x

-intercept is the input value for which the output of the function is zero, so we solve

f(x) = 0

.

3

Find the

x

-intercept of the graph of

g

x = -\frac{3k + 2}{k}

We set the equation of

g(x)

to zero and solve for

x

in terms of

k

.

4

Set up and solve the equation relating the two

x

-intercepts

k = 6

Using the relation

x_g = 5 \cdot x_f

, we obtain

-\frac{3k + 2}{k} = 5 \left(-\frac{4}{k}\right)

, which simplifies to

3k + 2 = 20

, leading to

k = 6

.

Key Concept

Function translations and finding intercepts of linear equations

Question 118Question →

In the $xy$ -plane, a line with a positive slope $m$ and a $y$ -intercept of $(0, b)$ passes through the point $(b, 12)$ , where $b > 0$ . If the area of the triangle bounded by the line, the $x$ -axis, and the $y$ -axis is $4$ , what is the value of $m$ ?

Show answer & explanation

Answer: 2

Answer

2

The line equation is

y = mx + b

. Substituting the point

(b, 12)

gives

12 = b(m + 1)

. The

x

-intercept of the line is at

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

, which gives a base length of

\frac{b}{m}

for the right triangle, while the height is

b

. The area of the triangle is

\frac{b^2}{2m} = 4 \implies b^2 = 8m

. Substituting

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

into this equation yields

\frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Solving this for positive values of

m

gives the unique solution

m = 2

.

Step-by-Step Solution

1

Express the line equation and substitute the given point

(b, 12)

.

b(m + 1) = 12

Since the line has a

y

-intercept of

(0, b)

and a slope of

m

, its equation is

y = mx + b

. Substituting the point

(b, 12)

yields

12 = m(b) + b = b(m + 1)

.

2

Determine the lengths of the base and height of the triangle formed by the line and the coordinate axes.

Base

= \frac{b}{m}

, Height

= b

The

y

-intercept is at

(0, b)

and the

x

-intercept is at

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

. Since

b > 0

and

m > 0

, the height is

b

and the base is the absolute value of the

x

-intercept, which is

\frac{b}{m}

.

3

Write the area equation of the triangle and simplify it.

b^2 = 8m

The area of the right triangle is

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

. Setting this equal to the given area of

4

gives

\frac{b^2}{2m} = 4 \implies b^2 = 8m

.

4

Solve the system of equations for

m

.

m = 2

Substituting

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

into

b^2 = 8m

yields

\left(\frac{12}{m + 1}\right)^2 = 8m \implies \frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Since

m(m + 1)^2

is strictly increasing for positive

m

, testing small integers shows

2(3)^2 = 18

, so

m = 2

is the unique positive real solution.

Key Concept

Formulating linear equations in slope-intercept form and solving non-linear systems of equations derived from geometric constraints.

Alternative Method

Instead of algebraically expanding the cubic equation, one can test small positive integers for

m

in the equation

m(m + 1)^2 = 18

. Testing

m = 1

yields

1(2)^2 = 4

(too small), and testing

m = 2

yields

2(3)^2 = 18

(correct). Because

m(m + 1)^2

is strictly increasing for

m > 0

,

m = 2

is the only positive real root.

Estimated Time:3m 0s

Question 119Question →

If $-2(x - 5) \geq 4$ , what is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 3

Answer

3

Distributing the

-2

on the left side of the inequality

-2(x - 5) \geq 4

yields

-2x + 10 \geq 4

. Subtracting

10

from both sides gives

-2x \geq -6

. Dividing both sides by

-2

and reversing the inequality sign results in

x \leq 3

. The maximum possible value is therefore 3.

Step-by-Step Solution

1

Distribute

-2

to the terms inside the parentheses.

-2x + 10 \geq 4

Applying the distributive property simplifies the inequality.

2

Subtract

10

from both sides of the inequality.

-2x \geq -6

This isolates the variable term on the left side.

3

Divide both sides of the inequality by

-2

and reverse the inequality sign.

x \leq 3

Dividing both sides of an inequality by a negative number reverses the direction of the inequality.

Key Concept

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

Question 120Question →

A company manufactures custom travel mugs. The daily cost $C$ , in dollars, to manufacture $n$ mugs is given by the formula $C = 1.25n + 250$ . The company sells each mug for $4.50. To make a profit of exactly$ 400$ in a day, how many mugs must the company manufacture?

Show answer & explanation

Answer: 200

Answer

The company must manufacture 200 mugs to make a profit of exactly $400 in a day.

Subtracting the cost expression from the revenue expression gives the profit equation:

4.50n - (1.25n + 250) = 400

. Distributing the negative sign results in

4.50n - 1.25n - 250 = 400

, which simplifies to

3.25n - 250 = 400

. Adding 250 to both sides gives

3.25n = 650

. Dividing both sides by 3.25 yields

n = 200

.

Step-by-Step Solution

1

Set up the profit equation

4.50n - (1.25n + 250) = 400

Profit is calculated as total revenue (

4.50n

) minus total cost (

1.25n + 250

).

2

Distribute the negative sign and simplify the equation

3.25n - 250 = 400

Subtracting the entire cost expression requires distributing the subtraction to both

1.25n

and

250

, yielding

4.50n - 1.25n - 250

.

3

Isolate the variable term by adding 250 to both sides

3.25n = 650

Adding 250 to both sides of the equation isolates the term containing

n

.

4

Solve for

n

by dividing both sides by 3.25

n = 200

Dividing 650 by 3.25 gives the number of mugs needed to achieve the target profit.

Key Concept

Solving linear equations in one variable derived from a real-world context.

Algebra

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