Algebra Practice Questions — SAT — Page 9

Question 161Question →

A graphic designer is exporting images for a website. There are two types of images: standard images and high-resolution images. Each standard image, $x$ , requires 3 megabytes (MB) of storage and takes 2 seconds to upload. Each high-resolution image, $y$ , requires 8 MB of storage and takes 5 seconds to upload. The designer must limit the total storage of the exported images to at most 120 MB and the total upload time to at least 60 seconds. Which of the following systems of inequalities models this situation?

3x + 2y \leq 120

8x + 5y \geq 60

3x + 8y \geq 120

2x + 5y \leq 60

8x + 3y \leq 120

5x + 2y \geq 60

3x + 8y \leq 120

2x + 5y \geq 60

Show answer & explanation

Answer: $3x + 8y \leq 120$
$2x + 5y \geq 60$

Answer

The system of inequalities is

3x + 8y \leq 120

and

2x + 5y \geq 60

.

The correct system models the limits correctly. The storage constraint is represented by

3x + 8y \leq 120

since 'at most' corresponds to a less-than-or-equal-to sign. The upload time constraint is represented by

2x + 5y \geq 60

since 'at least' corresponds to a greater-than-or-equal-to sign.

Step-by-Step Solution

1

Set up the inequality for the storage constraint.

3x + 8y \leq 120

Each standard image uses 3 megabytes and each high-resolution image uses 8 megabytes. The total storage must be at most (less than or equal to) 120 megabytes.

2

Set up the inequality for the upload time constraint.

2x + 5y \geq 60

Each standard image takes 2 seconds and each high-resolution image takes 5 seconds to upload. The total upload time must be at least (greater than or equal to) 60 seconds.

3

Combine both inequalities into a system.

The system containing

3x + 8y \leq 120

and

2x + 5y \geq 60

.

Both constraints must be satisfied simultaneously.

Key Concept

Translating verbal constraints into systems of linear inequalities in two variables.

Question 162Question →

In the $xy$ -plane, a system of two linear equations has no solutions. One of the equations in the system is $4x - 6y = 15$ . The graph of the second equation is a line that passes through the points $(3, k)$ and $(9, 7)$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of

k

is

3

.

For a system of linear equations to have no solutions, the lines representing the equations must be parallel, which means they have equal slopes but different y-intercepts. The first equation,

4x - 6y = 15

, can be rewritten in slope-intercept form as

y = \frac{2}{3}x - 2.5

, showing its slope is

\frac{2}{3}

. The slope of the second line, passing through

(3, k)

and

(9, 7)

, is given by

\frac{7 - k}{9 - 3} = \frac{7 - k}{6}

. Setting the two slopes equal yields

\frac{2}{3} = \frac{7 - k}{6}

. Multiplying both sides by

6

gives

4 = 7 - k

, which solves to

k = 3

. Substituting

k = 3

back into the second line gives

y = \frac{2}{3}x + 1

. Since the slopes are equal and the y-intercepts (

-2.5

and

1

) are different, the lines are parallel and distinct, confirming there are no solutions.

Step-by-Step Solution

1

Find the slope of the first line by converting the equation to slope-intercept form.

Slope is

\frac{2}{3}

and y-intercept is

-2.5

.

Converting

4x - 6y = 15

to

y = \frac{2}{3}x - 2.5

reveals the slope of the first line.

2

Express the slope of the second line using the coordinates of the two points on the line.

Slope expression is

\frac{7 - k}{6}

.

Applying the slope formula

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

to

(3, k)

and

(9, 7)

defines the slope in terms of

k

.

3

Equate the two slopes and solve for the unknown parameter.

k = 3

Since the system has no solutions, the lines must be parallel and have equal slopes. Setting

\frac{2}{3} = \frac{7 - k}{6}

and solving gives

k = 3

.

Key Concept

For a system of two linear equations to have no solutions, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts.

Estimated Time:2m 0s

Question 163Question →

If $3x - 2(y - 5) = 12$ and $x = 8$ , what is the value of $y$ ?

1

6

11

10

Show answer & explanation

Answer: 11

Answer

11

The correct answer is 11. Substituting

x = 8

into the equation gives

3(8) - 2(y - 5) = 12

, which simplifies to

24 - 2y + 10 = 12

. Combining the constants on the left side yields

34 - 2y = 12

. Subtracting 34 from both sides gives

-2y = -22

. Finally, dividing both sides by

-2

yields

y = 11

.

Step-by-Step Solution

1

Substitute

x = 8

into the given equation.

3(8) - 2(y - 5) = 12

, which simplifies to

24 - 2(y - 5) = 12

.

This isolates the variable

y

by replacing the variable

x

with its given value.

2

Distribute

-2

to the terms inside the parentheses.

24 - 2y + 10 = 12

.

Multiplying

-2

by

y

gives

-2y

, and multiplying

-2

by

-5

gives

+10

because the product of two negative numbers is positive.

3

Combine the constant terms on the left side of the equation and subtract the result from both sides.

34 - 2y = 12 \Rightarrow -2y = -22

.

Combining

24

and

10

gives

34

. Subtracting

34

from both sides isolates the variable term

-2y

.

4

Divide both sides by

-2

to solve for

y

.

y = 11

.

Dividing both sides of the equation by the coefficient of the variable term isolates the variable.

Key Concept

Solving linear equations in two variables by substitution and isolation.

Estimated Time:45s

Question 164Question →

Which of the following represents all possible values of $x$ that satisfy the inequality $-2(x - 5) > 6$ ?

x < -8

x < 2

x > 2

x < 8

Show answer & explanation

Answer: $x < 2$

Answer

The inequality is satisfied when

x < 2

.

Dividing both sides of the inequality

-2(x - 5) > 6

by

-2

yields

x - 5 < -3

, where the inequality sign is reversed because of division by a negative number. Adding

5

to both sides of

x - 5 < -3

results in

x < 2

.

Step-by-Step Solution

1

Divide both sides of the inequality

-2(x - 5) > 6

by

-2

.

x - 5 < -3

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

2

Add

5

to both sides of the inequality to isolate

x

.

x < 2

To solve for

x

, eliminate the constant term

-5

on the left side of the inequality.

Key Concept

Solving linear inequalities in one variable, including reversing the inequality sign when multiplying or dividing by a negative number.

Question 165Question →

A system of inequalities is shown below.

\begin{aligned} y &\leq -3x + 15 \\ y &\leq 2x + 2 \end{aligned}

If

(x, y)

is a solution to the system, what is the maximum possible integer value of

y

?

6

7

8

15

Show answer & explanation

Answer: 7

Answer

7

The maximum value of

y

in the system occurs at the intersection of the boundary lines

y = -3x + 15

and

y = 2x + 2

. Solving for the intersection gives

x = 2.6

and

y = 7.2

. Since any point in the solution set must satisfy

y \leq 7.2

, the largest possible integer value for

y

is 7. We can confirm this is achievable because when

y = 7

,

x

can be any value in the range

[2.5, 2.67]

(such as

x=2.6

), which satisfies both inequalities.

Step-by-Step Solution

1

Find the intersection point of the boundary lines.

Set the two equations equal to find the

x

-coordinate:

-3x + 15 = 2x + 2 \implies 5x = 13 \implies x = 2.6

.

The maximum value of

y

in a system bounded from above by two lines with opposing slopes occurs at their point of intersection.

2

Calculate the corresponding maximum

y

-value.

Substitute

x = 2.6

into either boundary equation:

y = 2(2.6) + 2 = 7.2

.

This determines the absolute upper bound of

y

for any coordinate pair in the solution set.

3

Determine the maximum integer value of

y

.

Since

y \leq 7.2

, the largest integer value that

y

can take is 7.

The question asks specifically for the maximum integer value of

y

, and since

7 \leq 7.2

, a solution exists at this

y

-value.

4

Verify that a solution exists for

y = 7

.

For

y = 7

, the system requires

7 \leq -3x + 15 \implies x \leq 2.67

and

7 \leq 2x + 2 \implies x \geq 2.5

. The interval

[2.5, 2.67]

is non-empty, so valid solutions (e.g.,

(2.6, 7)

) exist.

This confirms that

y = 7

is attainable within the feasible region.

Key Concept

Optimization of variables within systems of linear inequalities

Question 166Question →

A graphic designer allocates their weekly work time between designing layouts, which takes $d$ hours, and client consultations, which takes $c$ hours. The designer works a maximum of 35 hours per week. Additionally, the designer spends at least twice as much time designing layouts as they do in client consultations. Which of the following systems of inequalities represents this situation?

\begin{aligned} d + c &\leq 35 \\ d &\geq 2c \end{aligned}

\begin{aligned} d + c &\geq 35 \\ d &\leq 2c \end{aligned}

\begin{aligned} d + c &\leq 35 \\ c &\geq 2d \end{aligned}

\begin{aligned} d + c &\geq 35 \\ c &\leq 2d \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} d + c &\leq 35 \\ d &\geq 2c \end{aligned}$

Answer

The system with the inequalities

d + c \leq 35

and

d \geq 2c

The system containing the inequalities

d + c \leq 35

and

d \geq 2c

is correct. The constraint 'works a maximum of 35 hours per week' means the sum of the hours,

d + c

, must be less than or equal to 35 (

d + c \leq 35

). The constraint 'spends at least twice as much time designing layouts as they do in client consultations' means that design hours,

d

, must be greater than or equal to twice the consultation hours,

c

(

d \geq 2c

).

Step-by-Step Solution

1

Identify the inequality representing the limit on total weekly hours.

d + c \leq 35

The phrase 'works a maximum of 35 hours per week' means the total hours spent on designing (

d

) and consulting (

c

) cannot exceed 35.

2

Identify the inequality representing the relative relationship between designing and consulting hours.

d \geq 2c

The phrase 'at least twice as much time designing layouts as they do in client consultations' means the design hours (

d

) must be greater than or equal to two times the consultation hours (

2c

).

3

Combine the individual inequalities into a system.

The system consists of

d + c \leq 35

and

d \geq 2c

.

Both conditions must be satisfied simultaneously to represent the designer's weekly schedule constraint.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 0s

Question 167Question →

A closed triangular region in the coordinate plane is defined by the following system of linear inequalities:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ x &\geq 1 \end{aligned}

What is the maximum possible value of the expression $2x + y$ for any point $(x, y)$ that lies within or on the boundary of this region?

Show answer & explanation

Answer: 12

Answer

The maximum possible value of the expression

2x + y

is 12.

To find the maximum possible value of the expression

2x + y

subject to the given system of inequalities, we first identify the boundary lines and find the vertices of the bounded triangular region in the coordinate plane. The boundary lines are

y = 2x - 4

,

y = -x + 8

, and

x = 1

. The intersection of

y = 2x - 4

and

y = -x + 8

occurs at

x = 4, y = 4

, which gives vertex

(4, 4)

. The intersection of

y = 2x - 4

and

x = 1

occurs at

(1, -2)

. The intersection of

y = -x + 8

and

x = 1

occurs at

(1, 7)

. Evaluating the linear expression

2x + y

at these three vertices gives

2(4) + 4 = 12

,

2(1) - 2 = 0

, and

2(1) + 7 = 9

. By the corner point theorem, the maximum value of a linear function on a closed polygonal region occurs at one of the vertices. Comparing the values, the maximum possible value is 12.

Step-by-Step Solution

1

Find the vertex formed by the intersection of the boundary lines

y = 2x - 4

and

y = -x + 8

.

Vertex

A(4, 4)

Setting the two equations equal:

2x - 4 = -x + 8 \implies 3x = 12 \implies x = 4

. Substituting

x = 4

back into either equation yields

y = 4

.

2

Find the vertex formed by the intersection of the boundary line

y = 2x - 4

and the vertical line

x = 1

.

Vertex

B(1, -2)

Substituting

x = 1

into

y = 2x - 4

gives

y = 2(1) - 4 = -2

.

3

Find the vertex formed by the intersection of the boundary line

y = -x + 8

and the vertical line

x = 1

.

Vertex

C(1, 7)

Substituting

x = 1

into

y = -x + 8

gives

y = -1 + 8 = 7

.

4

Evaluate the expression

2x + y

at each of the three vertices.

At

A(4, 4)

, the value is

12

; at

B(1, -2)

, the value is

0

; at

C(1, 7)

, the value is

9

.

According to the Corner Point Theorem of linear programming, the maximum or minimum of a linear objective function on a closed bounded region must occur at one of the vertices.

5

Identify the maximum value from the evaluated points.

The maximum value is 12.

Comparing the values 12, 0, and 9 shows that 12 is the largest value.

Key Concept

Linear Programming and Systems of Inequalities

Question 168Question →

An online streaming service offers the two monthly subscription plans described in the table below:

Plan	Monthly fee	Cost per premium movie rental
Plan A	$12	$1.50
Plan B	$27 (includes first 4 rentals)	$0.75 (for each rental after the first 4)

If a user rented $m$ premium movies in a month, where $m > 4$ , and the total cost for both plans would be the same, what is the value of $m$ ?

Show answer & explanation

Answer: 16

Answer

The value of

m

that results in the same total monthly cost for both plans is 16.

The correct answer of 16 represents the exact number of premium movie rentals where the total monthly cost for both plans is equal to $36. Any other number of movie rentals will result in different costs for the two plans.

Step-by-Step Solution

1

Define the cost equation for Plan A

\text{Cost}_A = 12 + 1.50m

Plan A charges a flat

12 monthly fee plus

1.50 for each of the

m

movies rented.

2

Define the cost equation for Plan B

\text{Cost}_B = 27 + 0.75(m - 4)

Plan B charges a

27 monthly fee that covers the first 4 movies, and

0.75 for each of the

m - 4

additional movies rented because

m > 4

.

3

Equate the two cost expressions and solve for

m

12 + 1.50m = 27 + 0.75(m - 4) \implies 12 + 1.50m = 24 + 0.75m \implies 0.75m = 12 \implies m = 16

To find when the costs are identical, set the two algebraic expressions equal to each other and isolate the variable

m

.

Key Concept

Setting up and solving linear equations in one variable from context

Alternative Method

Instead of setting up full equations, we can look at the cost difference at

m = 4

. At

4

movies, Plan A costs

12 + 1.50(4) = 18

dollars, and Plan B costs

27

dollars (since

4

movies are included). The price difference is

27 - 18 = 9

dollars. For each movie rented beyond

4

, the cost of Plan A increases by

1.50

dollars while Plan B only increases by

0.75

dollars. The rate of change difference is

1.50 - 0.75 = 0.75

dollars per movie. To bridge the initial

9

dollar difference, the user needs to rent

\frac{9}{0.75} = 12

more movies. Thus, the total number of movies is

4 + 12 = 16

.

Estimated Time:1m 30s

Question 169Question →

The table below shows some values for a linear function $f$ , where $k$ is a positive constant.

$x$	$f(x)$
$0$	$3$
$k$	$2k + 6$
$4k$	$27$

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

2

-4

4

\frac{1}{4}

Show answer & explanation

Answer: 4

Answer

The slope of the graph of

f

is

4

.

The correct answer is

4

. A linear function can be written in the form

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-intercept. Since the

y

-intercept of the graph of

f

is

(0, 3)

, we have

b = 3

, so the function is

f(x) = mx + 3

. Using the given points, we can set up the equations

f(k) = mk + 3 = 2k + 6

and

f(4k) = 4mk + 3 = 27

. Solving this system yields

mk = 6

and

2k = 3

, which gives

k = 1.5

. Substituting

k = 1.5

into

mk = 6

gives the slope

m = 4

.

Step-by-Step Solution

1

Express the linear function

f(x)

using the slope-intercept form and the given

y

-intercept.

f(x) = mx + 3

, where

m

is the slope and the

y

-intercept is

(0, 3)

.

A linear function has the general form

f(x) = mx + b

. The table indicates that when

x = 0

,

f(x) = 3

, which gives the

y

-intercept

(0, 3)

and establishes

b = 3

.

2

Use the point

(k, 2k + 6)

from the table to write an equation involving

m

and

k

.

mk + 3 = 2k + 6 \implies mk = 2k + 3

.

Substituting

x = k

into the function expression gives

f(k) = mk + 3

. Setting this equal to the table value

2k + 6

allows us to express

mk

in terms of

k

.

3

Use the point

(4k, 27)

from the table to write another equation involving

m

and

k

, and solve for the product

mk

.

4mk + 3 = 27 \implies 4mk = 24 \implies mk = 6

.

Substituting

x = 4k

into the function expression gives

f(4k) = 4mk + 3

. Setting this equal to the table value

27

allows us to solve directly for the numerical value of

mk

.

4

Substitute the value of

mk

into the equation from Step 2 to solve for the constant

k

.

6 = 2k + 3 \implies 2k = 3 \implies k = 1.5

.

By replacing

mk

with

6

in the equation

mk = 2k + 3

, we get a single-variable linear equation that we can solve for

k

.

5

Solve for the slope

m

using the value of

k

and the product

mk

.

m(1.5) = 6 \implies m = 4

.

Since

mk = 6

and

k = 1.5

, dividing the product

6

by

1.5

yields the slope

m

.

Key Concept

Using coordinate points and intercepts to determine the slope of a linear function represented in a table.

Question 170Question →

A system of linear equations is given by

\begin{aligned} kx - 4y &= 12 \\ 3x + ky &= 5 \end{aligned}

where

k

is a constant. If the system has a unique solution

(x, y)

such that

x > 0

and

y < 0

, how many possible integer values of

k

exist?

7

8

9

10

Show answer & explanation

Answer: 9

Answer

9

To find the number of integer values of

k

for which the system has a solution with

x > 0

and

y < 0

, we first express

x

and

y

in terms of

k

. Eliminating

y

by multiplying the first equation by

k

and the second by

4

and adding them yields

(k^2 + 12)x = 12k + 20

, or

x = \frac{12k+20}{k^2+12}

. Similarly, eliminating

x

yields

y = \frac{5k-36}{k^2+12}

. Since the denominator

k^2 + 12

is strictly positive for all real

k

, the sign of

x

and

y

depends solely on their numerators. For

x > 0

, we require

12k + 20 > 0

, which gives

k > -\frac{5}{3} \approx -1.67

. For

y < 0

, we require

5k - 36 < 0

, which gives

k < \frac{36}{5} = 7.2

. Combining these constraints gives the interval

-\frac{5}{3} < k < \frac{36}{5}

. The integers in this interval are

-1, 0, 1, 2, 3, 4, 5, 6,

and

7

, which is a total of 9 integers.

Step-by-Step Solution

1

Solve the system of equations for

x

in terms of

k

by eliminating

y

.

x = \frac{12k + 20}{k^2 + 12}

Multiply the first equation by

k

and the second by 4, then add them:

k(kx - 4y) + 4(3x + ky) = 12k + 20 \implies (k^2 + 12)x = 12k + 20

.

2

Solve the system of equations for

y

in terms of

k

by eliminating

x

.

y = \frac{5k - 36}{k^2 + 12}

Multiply the first equation by 3 and the second by

k

, then subtract the first from the second:

k(3x + ky) - 3(kx - 4y) = 5k - 36 \implies (k^2 + 12)y = 5k - 36

.

3

Apply the condition

x > 0

to find a constraint on

k

.

k > -\frac{5}{3}

Since the denominator

k^2 + 12

is positive for all real values of

k

, the expression for

x

is positive if and only if its numerator is positive:

12k + 20 > 0 \implies k > -\frac{20}{12} = -\frac{5}{3}

.

4

Apply the condition

y < 0

to find another constraint on

k

.

k < \frac{36}{5}

Similarly, since

k^2 + 12 > 0

, the expression for

y

is negative if and only if its numerator is negative:

5k - 36 < 0 \implies k < \frac{36}{5}

.

5

Combine the constraints and count the number of integer values of

k

in the resulting interval.

9 integer values

The combined inequality is

-\frac{5}{3} < k < \frac{36}{5}

, which simplifies to approximately

-1.67 < k < 7.2

. The integers in this range are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, giving a total of 9 integers.

Key Concept

Solving systems of linear equations with parameters under inequality constraints

Question 171Question →

In the $xy$ -plane, a line passes through the points $(2, 9)$ and $(5, 21)$ . 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 the points

(x_1, y_1)

and

(x_2, y_2)

is determined by the formula

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

. Substituting the given points

(2, 9)

and

(5, 21)

into the formula gives

m = \frac{21 - 9}{5 - 2} = \frac{12}{3} = 4

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (2, 9)

and

(x_2, y_2) = (5, 21)

These points are used in the slope formula.

2

Apply the slope formula

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

to set up the calculation.

m = \frac{21 - 9}{5 - 2}

The slope is defined as the change in

y

divided by the change in

x

.

3

Simplify the fraction to calculate the final slope.

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

Dividing the vertical change by the horizontal change yields the slope.

Key Concept

Calculating the slope of a line given two points in the coordinate plane.

Question 172Question →

In the $xy$ -plane, the solution set to the system of inequalities below is a bounded region.

\begin{aligned} y &\leq -2x + 21 \\ y &\leq -\frac{1}{2}x + 9 \\ y &\geq -x + 5 \\ x &\geq 2 \\ y &\geq 1 \end{aligned}

What is the maximum possible value of $3x + 2y$ for a point $(x, y)$ in this region?

22

31

32

34

Show answer & explanation

Answer: 34

Answer

34

The correct value is 34. The solution set to the system of inequalities is a bounded pentagonal region. Evaluating the expression

3x + 2y

at each vertex of this region yields:

12

at

(2, 3)

,

22

at

(2, 8)

,

34

at

(8, 5)

,

32

at

(10, 1)

, and

14

at

(4, 1)

. Comparing these values shows that the maximum value is 34.

Step-by-Step Solution

1

Understand the behavior of the objective function over the region

The maximum value of a linear expression

3x + 2y

over a closed, bounded polygonal region must occur at one of the vertices (corner points) of the region.

This is a fundamental theorem of linear programming, which simplifies the search for the maximum value to only the boundary intersections of the system.

2

Determine the vertices of the feasible region

The boundary lines are

y = -2x + 21

,

y = -\frac{1}{2}x + 9

,

y = -x + 5

,

x = 2

, and

y = 1

. Finding their intersections that satisfy all inequalities yields five vertices:

(2, 3)

,

(2, 8)

,

(8, 5)

,

(10, 1)

, and

(4, 1)

.

Solving the pairwise equations of lines that bound the shaded region determines the exact coordinates of all corner points.

3

Evaluate the expression

3x + 2y

at each of the five vertices

At

(2, 3)

:

3(2) + 2(3) = 12

At

(2, 8)

:

3(2) + 2(8) = 22

At

(8, 5)

:

3(8) + 2(5) = 34

At

(10, 1)

:

3(10) + 2(1) = 32

At

(4, 1)

:

3(4) + 2(1) = 14

Calculating the value at each candidate vertex allows us to compare and find the absolute maximum.

4

Compare the evaluated values

The maximum value is

34

, occurring at the vertex

(8, 5)

.

Comparing all calculated values shows that 34 is the largest possible value.

Key Concept

Linear Programming and Bounded Systems of Inequalities

Estimated Time:3m 0s

Question 173Question →

If $3(2y - 5) - 4(y - 2) = 11$ , what is the value of $3y + 2$ ?

8

9

29

53

Show answer & explanation

Answer: 29

Answer

29

The correct value is obtained by first distributing the terms, simplifying the equation to find the value of

y

, and then evaluating the expression

3y + 2

. Specifically, distributing yields

6y - 15 - 4y + 8 = 11

, which simplifies to

2y - 7 = 11

. Adding

7

to both sides gives

2y = 18

, so

y = 9

. Substituting

y = 9

into

3y + 2

gives

3(9) + 2 = 29

.

Step-by-Step Solution

1

Distribute the coefficients outside the parentheses.

6y - 15 - 4y + 8 = 11

To simplify the linear equation, we must first remove the parentheses by multiplying the terms inside by the factors outside.

2

Combine like terms on the left side of the equation.

2y - 7 = 11

Combining the variable terms (

6y - 4y = 2y

) and constant terms (

-15 + 8 = -7

) simplifies the equation.

3

Isolate the variable term by adding 7 to both sides, then divide by 2.

y = 9

Adding 7 gives

2y = 18

, and dividing by 2 isolates

y

.

4

Substitute the value of

y

into the expression

3y + 2

.

3(9) + 2 = 29

The question asks for the value of the expression

3y + 2

, not just the variable

y

.

Key Concept

Linear Equations in One Variable

Question 174Question →

The linear function $f$ is defined such that its graph in the $xy$ -plane is perpendicular to the line $3x - 4y = 20$ . The graph of another linear function, $g$ , is the result of shifting the graph of $f$ right by $5$ units and down by $4$ units. If $f(0) = 8$ and the graph of $g$ intersects the $x$ -axis at the point $(k, 0)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

The slope of the line

3x - 4y = 20

is

\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal,

-\frac{4}{3}

. With a

y

-intercept of

(0, 8)

, the function is

f(x) = -\frac{4}{3}x + 8

. Translating this function

5

units right and

4

units down yields

g(x) = f(x - 5) - 4 = -\frac{4}{3}(x - 5) + 4 = -\frac{4}{3}x + \frac{32}{3}

. Setting

g(k) = 0

to find the

x

-intercept gives

-\frac{4}{3}k + \frac{32}{3} = 0

, which solves to

k = 8

.

Step-by-Step Solution

1

Find the slope of the line

3x - 4y = 20

.

The slope is

\frac{3}{4}

.

Converting the line equation to slope-intercept form

y = mx + b

gives

y = \frac{3}{4}x - 5

, where the coefficient of

x

represents the slope.

2

Find the slope of

f

.

The slope of

f

is

-\frac{4}{3}

.

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

3

Write the function

f(x)

.

f(x) = -\frac{4}{3}x + 8

.

Since the

y

-intercept is given by

f(0) = 8

and the slope is

-\frac{4}{3}

, the slope-intercept form is

f(x) = -\frac{4}{3}x + 8

.

4

Determine the function

g(x)

by translating

f(x)

5

units right and

4

units down.

g(x) = -\frac{4}{3}x + \frac{32}{3}

.

A translation of

f(x)

right by

5

units and down by

4

units corresponds to

g(x) = f(x - 5) - 4

. Substituting

x-5

into

f(x)

gives

g(x) = -\frac{4}{3}(x - 5) + 8 - 4 = -\frac{4}{3}x + \frac{32}{3}

.

5

Set

g(k) = 0

and solve for

k

.

k = 8

.

The graph of

g

intersects the

x

-axis at

(k, 0)

, which means

g(k) = 0

. Solving

-\frac{4}{3}k + \frac{32}{3} = 0

yields

k = 8

.

Key Concept

Writing linear functions using perpendicular slopes and performing vertical and horizontal transformations.

Question 175Question →

In the

xy

-plane, a system of inequalities is defined as follows:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ y &\geq \frac{1}{2}x \end{aligned}

How many points

(x, y)

with integer coordinates satisfy this system of inequalities?

Show answer & explanation

Answer: 5

Answer

The total number of points with integer coordinates that satisfy the system is 5.

The system of inequalities defines a closed triangular region in the coordinate plane. Finding the vertices of this triangle gives the horizontal boundaries for

x

, which are

\frac{8}{3} \approx 2.67

and

\frac{16}{3} \approx 5.33

. The only integers in this range are

x = 3

,

x = 4

, and

x = 5

. Testing each of these integers in the inequalities shows that when

x = 3

,

y

can be

2, 3, 4,

or

5

(4 points); when

x = 4

,

y

can only be

4

(1 point); and when

x = 5

, there are no valid solutions. Adding these together gives a total of 5 points.

Step-by-Step Solution

1

Find the intersection points of the three boundary lines to define the vertices of the solution region.

The intersection of

y = 2x - 4

and

y = \frac{1}{2}x

is at

x = \frac{8}{3} \approx 2.67, y = \frac{4}{3} \approx 1.33

.
The intersection of

y = -x + 8

and

y = \frac{1}{2}x

is at

x = \frac{16}{3} \approx 5.33, y = \frac{8}{3} \approx 2.67

.
The intersection of

y = 2x - 4

and

y = -x + 8

is at

x = 4, y = 4

.

Finding the vertices determines the exact boundaries of the solution region in the coordinate plane.

2

Identify the integer

x

-coordinates that lie within the horizontal span of the region.

The

x

-coordinates of the region range from

\frac{8}{3} \approx 2.67

to

\frac{16}{3} \approx 5.33

. The integers in this interval are

x = 3

,

x = 4

, and

x = 5

.

Since both coordinates must be integers, we only need to test integer values of

x

that fall within the boundaries of the region.

3

Find the integer

y

-values for each candidate

x

-value.

For

x = 3

: The inequalities require

y \geq 2(3) - 4 = 2

,

y \leq -3 + 8 = 5

, and

y \geq \frac{1}{2}(3) = 1.5

. Thus,

2 \leq y \leq 5

. The integer solutions are

y = 2, 3, 4, 5

(4 points).
For

x = 4

: The inequalities require

y \geq 2(4) - 4 = 4

,

y \leq -4 + 8 = 4

, and

y \geq \frac{1}{2}(4) = 2

. Thus,

4 \leq y \leq 4

, which means

y = 4

(1 point).
For

x = 5

: The inequalities require

y \geq 2(5) - 4 = 6

and

y \leq -5 + 8 = 3

. No real number

y

can satisfy both

y \geq 6

and

y \leq 3

(0 points).

Evaluating the system at each candidate

x

-value reveals the set of matching integer

y

-values.

4

Sum the number of points found for each integer

x

-value.

4 points (when

x = 3

) + 1 point (when

x = 4

) + 0 points (when

x = 5

) = 5 points.

This yields the total number of integer coordinate pairs

(x, y)

that satisfy the system of inequalities.

Key Concept

Analyzing a bounded region defined by a system of linear inequalities to find discrete integer solutions (lattice points).

Alternative Method

Graph the three boundary lines on a grid:

y = 2x - 4

(solid line, shaded above),

y = -x + 8

(solid line, shaded below), and

y = \frac{1}{2}x

(solid line, shaded above). Identify the triangular intersection region on the grid and count the grid intersections (lattice points) that lie within or on the boundaries of this shaded triangle.

Estimated Time:2m 30s

Question 176Question →

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

\begin{aligned} ax + by &= 24 \\ 2x - 5y &= -7 \end{aligned}

If the system has the same unique solution $(x, y)$ for all values of $a$ and $b$ that satisfy the equation $4a + 3b = 24$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x + y

is

7

.

The correct answer is

7

. The solution to the system must satisfy

ax + by = 24

for any constants

a

and

b

that satisfy the constraint

4a + 3b = 24

. By matching the coefficients of

a

and

b

in both equations, we find

x = 4

and

y = 3

. To verify, we substitute these coordinates into the second equation:

2(4) - 5(3) = 8 - 15 = -7

, which is correct. The sum of the coordinates is

x + y = 4 + 3 = 7

.

Step-by-Step Solution

1

Analyze the constraint on the constants

a

and

b

given by

4a + 3b = 24

and relate it to the first equation

ax + by = 24

.

Since the first equation

ax + by = 24

must hold for all pairs of

(a, b)

satisfying

4a + 3b = 24

, the coefficients must correspond directly, meaning

x = 4

and

y = 3

.

This shows that

(4, 3)

is a point that lies on the line

ax + by = 24

regardless of the specific values of

a

and

b

, as long as they satisfy the given constraint.

2

Prove the uniqueness of the point

(4, 3)

by testing specific values for

(a, b)

that satisfy the constraint

4a + 3b = 24

.

If

b = 0

, then

4a = 24 \implies a = 6

. The equation becomes

6x = 24 \implies x = 4

. If

a = 0

, then

3b = 24 \implies b = 8

. The equation becomes

8y = 24 \implies y = 3

. This confirms

(x, y) = (4, 3)

is the unique point.

Choosing convenient test values for

a

and

b

helps isolate the variables

x

and

y

to verify the coordinates of the solution.

3

Substitute the point

(4, 3)

into the second equation of the system,

2x - 5y = -7

, to verify that it is consistent.

2(4) - 5(3) = 8 - 15 = -7

. Since this statement is true,

(4, 3)

is the consistent unique solution to the system of equations.

A solution to a system of equations must satisfy all equations in the system.

4

Find the requested value of

x + y

using the coordinates of the solution.

x + y = 4 + 3 = 7

.

The question asks for the sum of the coordinates of the solution.

Key Concept

Systems of linear equations with parameter constraints

Question 177Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the point $(2, 7)$ . If this graph is translated $3$ units to the right and $4$ units down, the resulting graph passes through the point $(4, 2)$ . What is the value of $b$ ?

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. Since the original graph passes through

(2, 7)

and undergoes a translation of

3

units to the right and

4

units down, the point

(2, 7)

is translated to

(2 + 3, 7 - 4) = (5, 3)

on the new graph. The translated graph also passes through

(4, 2)

. Using the slope formula on the two points

(5, 3)

and

(4, 2)

of the translated line gives a slope of

1

. Since a translation preserves the slope, the original line also has a slope of

1

. Substituting

m = 1

and

(2, 7)

into the original equation

y = mx + b

yields

7 = 1(2) + b

, which simplifies to

b = 5

.

Step-by-Step Solution

1

Determine a point on the translated graph by translating the given point

(2, 7)

on the original graph.

The point

(5, 3)

is on the translated graph.

Since the entire graph is translated

3

units to the right and

4

units down, every point on the original graph is translated by the same vector

(+3, -4)

.

2

Calculate the slope of the translated graph using the points

(5, 3)

and

(4, 2)

.

The slope

m = 1

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the change in y divided by the change in x.

3

Find the slope of the original graph.

The slope of the original graph is also

1

.

Translating a line horizontally and vertically shifts its position but does not alter its slope.

4

Substitute the slope

m = 1

and the point

(2, 7)

into the slope-intercept form of the original line,

y = mx + b

, to solve for

b

.

b = 5

.

This determines the value of the y-intercept of the original line.

Key Concept

Linear Equations in Two Variables

Question 178Question →

An online store sells small gift boxes for $10$ dollars each and large gift boxes for $15$ dollars each. A customer wants to buy a total of at most $10$ boxes. If the customer must spend at least $120$ dollars on the boxes, what is the minimum number of large gift boxes the customer must buy?

Show answer & explanation

Answer: 4

Answer

The minimum number of large gift boxes the customer must buy is 4.

Using the constraints from the problem, we set up the system of inequalities:

x + y \leq 10

and

10x + 15y \geq 120

. Expressing the first inequality as

x \leq 10 - y

and substituting it into the second gives

10(10 - y) + 15y \geq 120

. Simplifying this results in

100 + 5y \geq 120

, which simplifies to

5y \geq 20

, or

y \geq 4

. Therefore, the minimum number of large gift boxes the customer must buy is

4

.

Step-by-Step Solution

1

Define variables for the quantities of each box.

Let

x

be the number of small boxes and

y

be the number of large boxes.

Establishing variables is necessary to translate the word problem into algebraic inequalities.

2

Write the system of inequalities representing the constraints.

x + y \leq 10

and

10x + 15y \geq 120

The total number of boxes is at most

10

, and the total cost must be at least

120

dollars.

3

Express

x

in terms of

y

using the first inequality.

x \leq 10 - y

This allows substitution into the second inequality to solve for the target variable

y

.

4

Substitute

x \leq 10 - y

into the second inequality and simplify.

100 + 5y \geq 120

Solving the resulting single-variable inequality will determine the possible values of

y

.

5

Solve the inequality for

y

.

y \geq 4

Subtracting

100

and dividing by

5

isolates

y

to show its minimum possible value is

4

.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 179Question →

A theater sells adult tickets for $x$ dollars each and child tickets for $y$ dollars each. For a morning show, the theater sold 2 adult tickets and 1 child ticket, collecting a total of $14$ dollars. For an afternoon show, the theater sold 3 adult tickets and 2 child tickets, collecting a total of $23$ dollars. What is the cost, in dollars, of 1 adult ticket and 1 child ticket combined?

5

6

9

19

Show answer & explanation

Answer: 9

Answer

The combined cost of 1 adult ticket and 1 child ticket is 9 dollars.

The correct answer is 9. Subtracting the equation representing the morning show (

2x + y = 14

) from the equation representing the afternoon show (

3x + 2y = 23

) directly yields

x + y = 9

. Alternatively, solving the system of equations by substitution or elimination gives

x = 5

(cost of an adult ticket) and

y = 4

(cost of a child ticket), and adding these two values results in

9

.

Step-by-Step Solution

1

Write the system of linear equations based on the problem description.

The system of equations is:

\begin{aligned} 2x + y &= 14 \\ 3x + 2y &= 23 \end{aligned}

To mathematically represent the ticket sales and total revenue for the morning and afternoon shows.

2

Subtract the first equation from the second equation to find the value of

x + y

directly.

(3x + 2y) - (2x + y) = 23 - 14 \implies x + y = 9

Subtracting the equations isolates the expression

x + y

immediately, which represents the combined cost of 1 adult ticket and 1 child ticket.

Key Concept

Solving systems of linear equations by elimination or subtraction to find a linear combination of variables.

Alternative Method

Solve the system using elimination: multiply the first equation by 2 to get

4x + 2y = 28

. Subtract the second equation (

3x + 2y = 23

) from this new equation to find

x = 5

. Substitute

x = 5

into the first equation to find

y = 4

. Finally, add the values to get

x + y = 9

.

Estimated Time:1m 30s

Question 180Question →

A sports store orders standard helmets for $20$ dollars each and premium helmets for $50$ dollars each. The store can spend at most $2{}200$ dollars on this order. The distributor requires the store to order at least $60$ helmets in total. If the store decides to order at least $15$ premium helmets, what is the maximum number of standard helmets that the store can order?

Show answer & explanation

Answer: 72

Answer

The maximum number of standard helmets the store can order is 72.

By setting the number of premium helmets to its minimum allowed value of

15

to maximize the budget remaining for standard helmets, we find

20x \leq 1450

, which yields

x \leq 72.5

. Since standard helmets must be ordered in whole numbers, the maximum number is

72

, which also satisfies the minimum total order of

60

helmets (

72 + 15 = 87 \geq 60

).

Step-by-Step Solution

1

Define variables and translate constraints into inequalities

Let

x

be the number of standard helmets and

y

be the number of premium helmets. The constraints are

20x + 50y \leq 2200

,

x + y \geq 60

, and

y \geq 15

.

To represent the problem's mathematical relationships using a system of linear inequalities.

2

Solve for the upper limit of

x

using the budget constraint and the minimum value of

y

Since

20x \leq 2200 - 50y

,

x

is maximized when

y

is at its minimum value,

y = 15

. Substituting

y = 15

yields

20x + 750 \leq 2200 \implies 20x \leq 1450 \implies x \leq 72.5

.

To find the maximum possible value of standard helmets under the budget constraint.

3

Verify with the minimum total order constraint and determine the maximum integer value

Checking

x + y \geq 60

with

y = 15

gives

x + 15 \geq 60 \implies x \geq 45

. Since

x

must satisfy

45 \leq x \leq 72.5

and must be an integer, the maximum integer value is

72

.

To ensure the solution is physically possible as a whole number of items and satisfies all constraints.

Key Concept

Systems of Linear Inequalities in Two Variables

Algebra

Answer

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