Algebra Practice Questions — SAT — Page 14

Consider the system of equations below.

\begin{aligned} 2x - y &= 4 \\ x + 2y &= 7 \end{aligned}

If

(x, y)

is the solution to the system of equations, what is the value of

x + y

?

1

5

8

11

Show answer & explanation

Answer: 5

Answer

The value of

x + y

is 5.

To solve the system, we can express

y

in terms of

x

from the first equation:

y = 2x - 4

. Substituting this into the second equation gives

x + 2(2x - 4) = 7

. Distributing the 2 gives

x + 4x - 8 = 7

, which simplifies to

5x - 8 = 7

. Adding 8 to both sides yields

5x = 15

, which gives

x = 3

. Substituting

x = 3

back into the first equation yields

2(3) - y = 4 \implies 6 - y = 4 \implies y = 2

. Thus, the value of

x + y

is

3 + 2 = 5

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the first equation.

y = 2x - 4

This sets up the system for solving by substitution.

2

Substitute the expression for

y

into the second equation and solve for

x

.

x + 2(2x - 4) = 7 \implies 5x - 8 = 7 \implies 5x = 15 \implies x = 3

This isolates and solves for the variable

x

.

3

Substitute the value of

x

back into the expression for

y

, then compute

x + y

.

y = 2(3) - 4 = 2

, and

x + y = 3 + 2 = 5

This finds the value of

y

and computes the final required sum.

Key Concept

Solving a system of linear equations in two variables to find the value of a linear combination.

Question 262Question →

An artist creates custom oil paintings and charcoal sketches.

* Each oil painting, $x$ , requires $4$ hours of setup time and $2$ hours of finishing time.
* Each charcoal sketch, $y$ , requires $1$ hour of setup time and $3$ hours of finishing time.

Each week, the artist can spend at most $20$ hours on setup time and at most $30$ hours on finishing time. Which of the following systems of inequalities represents this situation?

\begin{aligned} 4x + y &\ge 20 \\ 2x + 3y &\ge 30 \end{aligned}

\begin{aligned} 4x + 2y &\le 20 \\ x + 3y &\le 30 \end{aligned}

\begin{aligned} 4x + y &\le 20 \\ 2x + 3y &\le 30 \end{aligned}

\begin{aligned} 4x + y &\le 30 \\ 2x + 3y &\le 20 \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} 4x + y &\le 20 \\ 2x + 3y &\le 30 \end{aligned}$

Answer

The system of inequalities consisting of the inequalities

4x + y \le 20

and

2x + 3y \le 30

The variable

x

represents the number of oil paintings and

y

represents the number of charcoal sketches. Setup time constraint: each oil painting requires

4

hours and each charcoal sketch requires

1

hour. Thus, the total setup time is

4x + y

hours. Since the artist can spend at most

20

hours on setup time, this is represented by the inequality

4x + y \le 20

. Finishing time constraint: each oil painting requires

2

hours and each charcoal sketch requires

3

hours. Thus, the total finishing time is

2x + 3y

hours. Since the artist can spend at most

30

hours on finishing time, this is represented by the inequality

2x + 3y \le 30

. Combining these gives the correct system.

Step-by-Step Solution

1

Identify the variables and setup times for each type of artwork to write the setup time inequality.

The total setup time is

4x + y

hours. Since the limit is at most

20

hours, the inequality is

4x + y \le 20

.

To establish the mathematical representation of the setup time constraint.

2

Identify the variables and finishing times for each type of artwork to write the finishing time inequality.

The total finishing time is

2x + 3y

hours. Since the limit is at most

30

hours, the inequality is

2x + 3y \le 30

.

To establish the mathematical representation of the finishing time constraint.

3

Combine the two inequalities into a system.

The system of inequalities is

4x + y \le 20

and

2x + 3y \le 30

.

To find the system that represents both constraints simultaneously.

Key Concept

Translating real-world constraints into a system of linear inequalities in two variables.

Question 263Question →

An architect is tracking the height of a new skyscraper under construction. The total height $H$ , in feet, of the building $w$ weeks after construction of the main frames began is modeled by the equation $H = 120 + 15w$ . According to the model, by how many feet does the height of the building increase each week?

Show answer & explanation

Answer: 15

Answer

The height of the building increases by 15 feet each week.

In the linear model

H = 120 + 15w

, the variable

H

represents the total height of the building in feet and the variable

w

represents the number of weeks since construction began. The coefficient of the variable

w

, which is

15

, is the slope of the equation and represents the constant rate of change of the height per week. Therefore, the height of the building increases by

15

feet each week.

Step-by-Step Solution

1

Analyze the linear model equation.

The equation

H = 120 + 15w

represents the total height

H

as a function of the number of weeks

w

, where

15

is the coefficient of the variable

w

.

To determine what each part of the linear equation represents in the given context.

2

Interpret the coefficient of the independent variable

w

.

The coefficient of

w

is the slope of the linear equation, which is

15

.

The slope represents the constant rate of change, which is the weekly increase in height.

Key Concept

Interpreting the slope of a linear equation in context

Question 264Question →

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

\begin{aligned} kx - 3y &= 12 \\ (k-5)x - 2y &= 10 \end{aligned}

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

-15

-10

5

15

Show answer & explanation

Answer: 15

Answer

15

The correct answer is

15

. For a system of two linear equations to have no solution, the equations must represent parallel lines. This occurs when the coefficients of the variables are proportional but the constant terms are not. In this system, the ratio of the

x

-coefficients,

\frac{k}{k-5}

, must equal the ratio of the

y

-coefficients,

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

. Solving the equation

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

by cross-multiplying gives

2k = 3(k-5)

, which simplifies to

2k = 3k - 15

. Subtracting

3k

from both sides gives

-k = -15

, so

k = 15

. Since the ratio of the constants is

\frac{12}{10} = 1.2

, which is not equal to

1.5

, this value of

k

ensures the lines are parallel and distinct.

Step-by-Step Solution

1

Set up the condition for a system of linear equations to have no solution.

The coefficients of

x

and

y

must be proportional, but not equal to the ratio of the constant terms:

\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}

.

If the coefficients are proportional, the lines have the same slope. If the constant terms are not in the same proportion, the lines have different intercepts, meaning they are parallel and distinct, and thus never intersect.

2

Substitute the coefficients from the given equations into the proportion.

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

and verify that

\frac{-3}{-2} = 1.5 \neq \frac{12}{10} = 1.2

.

This establishes the relationship between the coefficients of

x

and

y

required to make the lines parallel, while ensuring they are not identical lines.

3

Solve the proportion for

k

.

\frac{k}{k-5} = \frac{3}{2} \Rightarrow 2k = 3(k-5) \Rightarrow 2k = 3k - 15 \Rightarrow k = 15

.

Cross-multiplying and isolating the variable

k

yields the value of the constant.

Key Concept

For a system of linear equations to have no solution, the lines represented by the equations must be parallel and distinct, meaning their slopes are equal but their y-intercepts are different.

Estimated Time:1m 30s

Question 265Question →

A subscription-based meal kit service charges a monthly membership fee of $\$35$ plus $\$8.50$ per meal. A non-member can purchase the same meals for $\$12.00$ each, but must pay a flat monthly delivery fee of $\$7$ . How many meals must be purchased in a month for the total monthly cost for a member to be equal to the total monthly cost for a non-member?

Show answer & explanation

Answer: 8

Answer

The total monthly cost is equal for a member and a non-member when 8 meals are purchased in a month.

The correct answer of 8 is found by setting the member cost expression,

35 + 8.5m

, equal to the non-member cost expression,

12m + 7

, and solving for the number of meals,

m

.

Step-by-Step Solution

1

Represent the total monthly cost for a member and a non-member using equations where

m

is the number of meals purchased.

Member cost is represented by

35 + 8.50m

, and non-member cost is represented by

12.00m + 7

.

To set up an algebraic representation of the cost structures.

2

Equate the two cost expressions to find the number of meals where the costs are equal.

35 + 8.50m = 12.00m + 7

To find the value of

m

that makes both costs equivalent.

3

Isolate the variable

m

on one side of the equation.

3.50m = 28

Subtracting

8.50m

and

7

from both sides simplifies the equation by keeping variable terms on one side and constants on the other.

4

Divide both sides of the equation by

3.50

to solve for

m

.

m = 8

Dividing

28

by

3.50

yields the final number of meals.

Key Concept

Solving linear equations in one variable with variable terms on both sides of the equation

Estimated Time:1m 30s

Question 266Question →

A bounded region in the $xy$ -plane is defined by the following system of inequalities:

\begin{aligned} y &\ge \frac{1}{2}x + 2 \\ y &\le 2x - 1 \\ y &\le -x + 11 \end{aligned}

What is the maximum $y$ -coordinate of any point within this region?

Show answer & explanation

Answer: 7

Answer

The maximum y-coordinate of any point in the bounded region is 7.

The solution region is a triangle with vertices at

(2, 3)

,

(4, 7)

, and

(6, 5)

. Since all inequalities are less-than-or-equal-to or greater-than-or-equal-to, the boundary points are included in the solution set. The maximum y-value occurs at the vertex

(4, 7)

, which gives a maximum y-coordinate of 7.

Step-by-Step Solution

1

Determine the equations of the boundary lines.

The boundary equations are

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

,

y = 2x - 1

, and

y = -x + 11

.

These equations represent the boundaries of the system of inequalities.

2

Find the intersection point of

y = 2x - 1

and

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

.

Solving

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

yields

x = 2

, which gives

y = 3

. The intersection is

(2, 3)

.

This is one vertex of the bounded region.

3

Find the intersection point of

y = 2x - 1

and

y = -x + 11

.

Solving

2x - 1 = -x + 11

yields

x = 4

, which gives

y = 7

. The intersection is

(4, 7)

.

This is the second vertex of the bounded region.

4

Find the intersection point of

y = -x + 11

and

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

.

Solving

-x + 11 = \frac{1}{2}x + 2

yields

x = 6

, which gives

y = 5

. The intersection is

(6, 5)

.

This is the third vertex of the bounded region.

5

Compare the y-coordinates of the three vertices of the bounded region.

The y-coordinates of the vertices

(2, 3)

,

(4, 7)

, and

(6, 5)

are 3, 7, and 5, respectively. The maximum value is 7.

The maximum value of a linear coordinate over a bounded convex region occurs at one of its vertices.

Key Concept

Finding the vertices of a bounded region defined by a system of linear inequalities and optimizing a coordinate value.

Question 267Question →

A botanist tracks the growth of a bamboo plant. The height $h$ , in inches, of the plant $d$ days after the tracking began can be modeled by the equation $h = 4.5d + 12$ . Which of the following is the best interpretation of $12$ in this context?

The height of the bamboo plant, in inches, 1 day after the tracking began.

The initial height of the bamboo plant, in inches, when the tracking began.

The average growth of the bamboo plant, in inches per day.

The number of days the tracking lasted.

Show answer & explanation

Answer: The initial height of the bamboo plant, in inches, when the tracking began.

Answer

The initial height of the bamboo plant, in inches, when the tracking began.

In the linear equation

h = 4.5d + 12

, the constant term

12

represents the y-intercept. This corresponds to the value of the function

h

when the independent variable

d

is equal to

0

. Since

d

is the number of days since tracking began,

d = 0

is the start of tracking, meaning

12

represents the initial height of the bamboo plant, in inches.

Step-by-Step Solution

1

Identify the component of the linear equation

h = 4.5d + 12

to be interpreted.

The number to interpret is the constant term

12

.

We must determine the contextual meaning of the constant value in the linear model.

2

Find the mathematical meaning of the constant in the slope-intercept form

y = mx + b

.

The constant term

12

represents the y-intercept of the linear equation, which occurs when the independent variable

d = 0

.

In a linear function, the constant represents the value of the dependent variable when the independent variable is zero.

3

Translate the mathematical definition to the real-world context of the problem.

Since

d

represents the number of days after tracking began,

d = 0

represents the start of tracking. At

d = 0

,

h = 12

, which is the initial height of the bamboo plant in inches.

Relating the y-intercept to the context yields the initial state of the measured quantity.

Key Concept

Interpreting the y-intercept of a linear model in context

Estimated Time:45s

Question 268Question →

A local coffee shop tracks its remaining syrup inventory at the end of each day. On day 3, the shop has 20 liters of syrup remaining. On day 7, the shop has 12 liters of syrup remaining. If the amount of syrup decreases at a constant daily rate, on which day will the shop have exactly 6 liters of syrup remaining?

4

7

10

31

Show answer & explanation

Answer: 10

Answer

10

To find the day when

6

liters of syrup remain, we first find the constant rate of decrease (slope) of the syrup. The slope is calculated as the change in syrup volume divided by the change in days:

m = \frac{12 - 20}{7 - 3} = -2

liters per day. Using the point-slope formula with the point

(3, 20)

, the relationship is

y - 20 = -2(x - 3)

, which simplifies to

y = -2x + 26

. Setting the remaining syrup

y

to

6

gives

6 = -2x + 26

. Solving for

x

yields

2x = 20

, so

x = 10

. Thus, the shop will have exactly

6

liters of syrup remaining on day

10

.

Step-by-Step Solution

1

Calculate the constant daily rate of decrease (slope) of the syrup inventory.

The rate of decrease is

-2

liters per day.

The slope

m

of a linear relationship between day

x

and remaining syrup

y

is given by

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 20}{7 - 3} = \frac{-8}{4} = -2

.

2

Determine the linear equation modeling the remaining syrup as a function of the day.

The linear equation is

y = -2x + 26

.

Using the point-slope form with the point

(3, 20)

and slope

m = -2

, we have

y - 20 = -2(x - 3)

. Distributing and simplifying gives

y - 20 = -2x + 6

, which simplifies to

y = -2x + 26

.

3

Find the day when the remaining syrup is exactly

6

liters.

The shop will have exactly

6

liters of syrup remaining on day

10

.

Substitute

y = 6

into the equation and solve for

x

:

6 = -2x + 26 \Rightarrow -20 = -2x \Rightarrow x = 10

.

Key Concept

Linear Functions and Graphs

Alternative Method

Instead of finding the full linear equation, we can use the rate of change directly. The syrup decreases by

2

liters per day. From day 7, where there are

12

liters remaining, we need to reach

6

liters remaining. This is a decrease of

12 - 6 = 6

liters. Since the rate of decrease is

2

liters per day, it will take

\frac{6}{2} = 3

additional days. Adding

3

days to day 7 gives

7 + 3 = 10

.

Estimated Time:1m 30s

Question 269Question →

A landscaping company is planting $x$ maple trees and $y$ pine trees in a park. The number of trees of each type must satisfy the system of inequalities below:

\begin{aligned} 30x + 40y &\le 360 \\ x + y &\ge 10 \\ x &\ge 4 \end{aligned}

What is the maximum number of pine trees the company can plant?

Show answer & explanation

Answer: 6

Answer

The maximum number of pine trees the company can plant is 6.

To find the maximum number of pine trees,

y

, we look at the boundary constraints. The constraint

x \ge 4

states that at least

4

maple trees must be planted. Since planting fewer maple trees leaves more of the budget for pine trees, we minimize

x

by setting

x = 4

. Substituting this value into the budget inequality

30x + 40y \le 360

gives

120 + 40y \le 360

. Solving for

y

yields

40y \le 240

, which simplifies to

y \le 6

. We then verify that the solution

(4, 6)

satisfies the total tree constraint

x + y \ge 10

, which it does since

4 + 6 = 10

. Thus, the maximum number of pine trees is 6.

Step-by-Step Solution

1

Substitute the minimum possible value of

x

into the first inequality.

Since

x \ge 4

, the smallest possible value for

x

is

4

. Substituting

x = 4

into

30x + 40y \le 360

yields:

30(4) + 40y \le 360

120 + 40y \le 360

To maximize the value of

y

under the resource constraint, we must minimize the value of

x

.

2

Solve the inequality for

y

.

40y \le 240

y \le 6

This establishes that the maximum possible value for

y

based on the budget constraint is

6

.

3

Verify that

(4, 6)

satisfies all inequalities in the system.

Checking the second inequality:

x + y \ge 10 \implies 4 + 6 = 10 \ge 10

, which is true. The third inequality

x \ge 4 \implies 4 \ge 4

is also true.

A coordinate pair must satisfy all inequalities in the system to be a valid solution.

Key Concept

To find the maximum value of a variable in a system of inequalities with constraints, analyze the boundary lines and the intersection points of the feasible region.

Question 270Question →

In the $xy$ -plane, the graph of a line $l$ passes through the points $(0, 1)$ and $(3, 5)$ . If another point on line $l$ has coordinates $(t, 9)$ , what is the value of $t$ ?

Show answer & explanation

Answer: 6

Answer

The value of

t

is

6

.

The slope of line

l

is

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

. Using the y-intercept

(0, 1)

, the equation of the line is

y = \frac{4}{3}x + 1

. Setting

y = 9

gives

9 = \frac{4}{3}t + 1

. Subtracting 1 from both sides gives

8 = \frac{4}{3}t

. Multiplying both sides by

\frac{3}{4}

yields

t = 6

.

Step-by-Step Solution

1

Calculate the slope of line

l

using the slope formula

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

with the points

(0, 1)

and

(3, 5)

.

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

The slope of a line represents its constant rate of change and is needed to determine the line's equation.

2

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

y = mx + b

, using the slope

m = \frac{4}{3}

and the y-intercept

b = 1

(from the point

(0, 1)

).

y = \frac{4}{3}x + 1

The slope-intercept equation defines the relationship between the

x

- and

y

-coordinates of any point on the line.

3

Substitute the point

(t, 9)

into the line's equation and solve for

t

.

9 = \frac{4}{3}t + 1 \implies 8 = \frac{4}{3}t \implies t = 6

Since the point lies on the line, its coordinates must satisfy the line's equation.

Key Concept

Determining the equation of a linear function from a graph or points and evaluating it for a given value.

Question 271Question →

A deep space communications satellite transmits a telemetry data file to a ground station on Earth. The remaining size of the file to be received at the ground station, $S$ , in megabytes (MB), can be modeled by the equation $S = 850 - 4.5(t - 18)$ , where $t$ is the number of seconds since the satellite initiated its transmission sequence, and $t \ge 18$ . Which of the following is the best interpretation of the number $18$ in this context?

The rate, in megabytes per second, at which the data is received by the ground station.

The size, in megabytes, of the telemetry file when the ground station begins receiving it.

The number of seconds after the transmission sequence is initiated before the ground station begins receiving the file.

The total number of seconds needed to fully transmit the telemetry file.

Show answer & explanation

Answer: The number of seconds after the transmission sequence is initiated before the ground station begins receiving the file.

Answer

The number of seconds after the transmission sequence is initiated before the ground station begins receiving the file.

The model

S = 850 - 4.5(t - 18)

is valid for

t \ge 18

. Substituting

t = 18

into the equation yields

S = 850

megabytes, which is the total size of the file before any data has been received. As

t

increases beyond

18

, the remaining file size decreases at a rate of

4.5

megabytes per second. Therefore, the first

18

seconds after the transmission sequence is initiated represent the time delay before the ground station starts receiving the file.

Step-by-Step Solution

1

Analyze the structure of the linear equation

S = 850 - 4.5(t - 18)

in context.

The variable

S

represents the remaining file size in megabytes, and

t

represents the time in seconds since the sequence was initiated. The model is defined only for

t \ge 18

.

Understanding the variables and constraints is the first step to interpreting the components of the equation.

2

Evaluate the equation at the boundary value

t = 18

.

When

t = 18

,

S = 850 - 4.5(18 - 18) = 850

megabytes.

This determines the starting state of the data reception modeled by the equation.

3

Analyze how

S

changes as

t

increases beyond

18

.

For every second

t

increases beyond

18

,

S

decreases by

4.5

megabytes.

This confirms that the transmission starts at

t = 18

seconds and proceeds at a rate of

4.5

megabytes per second, meaning the first

18

seconds represent the delay before data reception starts.

Key Concept

Interpreting Linear Relationships in Context

Question 272Question →

For the system of equations shown below,

(x, y)

is the unique solution:

\begin{aligned} \frac{3}{4}(2x - y) &= \frac{2}{3}(x + 2y) \\ \frac{1}{2}(x - y) + \frac{5}{6}(2x + y) &= 11.5 \end{aligned}

What is the value of

x + y

?

Show answer & explanation

Answer: 7

Answer

The value of

x + y

is 7.

To find the value of

x + y

, we first simplify the first equation by multiplying both sides by the least common multiple of

3

and

4

, which is

12

. This yields

9(2x - y) = 8(x + 2y)

, which simplifies to

18x - 9y = 8x + 16y

, and further simplifies to

10x = 25y

, or

x = 2.5y

. Next, we substitute

x = 2.5y

into the second equation:

\frac{1}{2}(2.5y - y) + \frac{5}{6}(2(2.5y) + y) = 11.5

. Simplifying the terms gives

0.75y + 5y = 11.5

, or

5.75y = 11.5

, which yields

y = 2

. Substituting

y = 2

back into

x = 2.5y

gives

x = 5

. Thus, the value of

x + y

is

5 + 2 = 7

.

Step-by-Step Solution

1

Clear the fractions in the first equation by multiplying by the least common multiple of the denominators.

10x = 25y

, which simplifies to

x = 2.5y

.

To express one variable in terms of the other for substitution.

2

Substitute the expression for

x

into the second equation and solve for

y

.

y = 2

To find the numerical value of one of the variables.

3

Substitute the value of

y

back into the simplified first equation to find

x

.

x = 5

To find the numerical value of the remaining variable.

4

Add the values of

x

and

y

to find

x + y

.

7

To answer the specific question asked.

Key Concept

Solving systems of linear equations using algebraic simplification and substitution

Question 273Question →

A landscaping service uses a water tank to irrigate lawns. The volume of water, $V$ , in gallons, remaining in the tank after $n$ lawns have been irrigated is modeled by the equation $V = 850 - 25n$ . According to the model, by how many gallons does the volume of water in the tank decrease for each lawn that is irrigated?

Show answer & explanation

Answer: 25

Answer

The correct answer is 25, which represents the decrease in the volume of water in the tank, in gallons, for each lawn irrigated.

In the linear equation

V = 850 - 25n

, the coefficient of

n

is

-25

. This coefficient represents the rate of change of the volume of water with respect to the number of lawns irrigated. The negative sign shows that the volume is decreasing, and the magnitude, 25, indicates that the volume decreases by 25 gallons for each lawn irrigated.

Step-by-Step Solution

1

Identify the coefficient of the variable

n

in the equation

V = 850 - 25n

.

The coefficient of

n

is

-25

.

The coefficient of the independent variable in a linear equation represents the rate of change of the dependent variable.

2

Interpret the coefficient in terms of the real-world context.

The coefficient

-25

means the volume of water decreases by 25 gallons for each lawn irrigated.

The negative sign indicates a decrease, and the magnitude represents the amount of change per unit.

Key Concept

Interpreting the slope of a linear relationship in context.

Question 274Question →

An environmental cleanup crew is removing a contaminant from a soil site. The remaining mass of the contaminant, $C$ , in kilograms, after $d$ days of treatment is modeled by the equation $C = 400 - p d$ , where $p$ is the daily removal rate, in kilograms per day, under the original protocol. Under a new treatment protocol, the daily removal rate is increased by $25\%$ , and the treatment time required to completely remove the contaminant is reduced by $8$ days. What was the daily removal rate, in kilograms per day, under the original protocol?

Show answer & explanation

Answer: 10

Answer

The daily removal rate under the original protocol was 10 kilograms per day.

Under the original protocol, complete removal of the

400

kg contaminant occurs when

C = 0

, giving a duration of

d = \frac{400}{p}

days. Under the new protocol, the removal rate increases by

25\%

to

1.25p

, and the duration is reduced by

8

days to

\frac{400}{p} - 8

. Because the total mass removed must still equal

400

kg, we write the equation

(1.25p)\left(\frac{400}{p} - 8\right) = 400

. Distributing

1.25p

yields

500 - 10p = 400

. Solving for

p

gives

10p = 100

, which simplifies to

p = 10

kilograms per day.

Step-by-Step Solution

1

Set

C = 0

in the original equation to represent complete removal.

0 = 400 - p d \implies d = \frac{400}{p}

Complete removal of the contaminant means that the remaining mass

C

is

0

kilograms.

2

Express the new daily removal rate and the new treatment duration using the given percentage increase and day reduction.

p_{\text{new}} = 1.25p

and

d_{\text{new}} = d - 8 = \frac{400}{p} - 8

The new protocol increases the daily removal rate by

25\%

and reduces the total treatment time by

8

days.

3

Set up the equation for complete removal under the new protocol using the new rate and duration.

400 - (1.25p) \left(\frac{400}{p} - 8\right) = 0

The total initial contaminant mass of

400

kilograms must be completely removed by the new daily rate over the new duration.

4

Solve the equation for

p

.

1.25p \left(\frac{400}{p} - 8\right) = 400 \implies 500 - 10p = 400 \implies 10p = 100 \implies p = 10

Distribute

1.25p

into the parentheses to eliminate the fraction, then isolate the variable

p

.

Key Concept

Interpreting the rate (slope) and intercepts of a linear relationship in context, and modeling variations of those parameters.

Question 275Question →

In a certain video game, players earn points for completing quests and defeating bosses. Completing a quest earns $q$ points, and defeating a boss earns $b$ points. Leo completed $5$ quests and defeated $3$ bosses, earning a total of $250$ points. Maya completed $7$ quests and defeated $2$ bosses, earning a total of $240$ points. What is the value of $q$ ?

Show answer & explanation

Answer: 20

Answer

The value of

q

is

20

.

To find the value of

q

, we translate the given scenarios into a system of two linear equations:

5q + 3b = 250

and

7q + 2b = 240

. Multiplying the first equation by

2

gives

10q + 6b = 500

. Multiplying the second equation by

3

gives

21q + 6b = 720

. Subtracting the first equation from the second yields

(21q - 10q) + (6b - 6b) = 720 - 500

, which simplifies to

11q = 220

. Dividing both sides by

11

results in

q = 20

.

Step-by-Step Solution

1

Set up the system of linear equations from the given information.

5q + 3b = 250

and

7q + 2b = 240

To represent the points earned by Leo and Maya mathematically.

2

Multiply the first equation by

2

and the second equation by

3

.

10q + 6b = 500

and

21q + 6b = 720

To make the coefficients of

b

equal so they can be eliminated.

3

Subtract the first modified equation from the second modified equation.

11q = 220

To eliminate

b

and solve for

q

directly.

4

Divide both sides of the equation by

11

.

q = 20

To find the number of points earned per completed quest.

Key Concept

Solving systems of two linear equations in two variables using elimination.

Question 276Question →

A laboratory technician is cooling a liquid sample. The initial temperature of the sample is $80^\circ\text{C}$ . The technician uses a cooling program that decreases the temperature at a constant rate of $1.5^\circ\text{C}$ per minute. After $t$ minutes, the technician increases the cooling rate by $0.75^\circ\text{C}$ per minute and runs the cooling program for another $12$ minutes. If the final temperature of the sample is $44^\circ\text{C}$ , what is the value of $t$ ?

8

6

18

42

Show answer & explanation

Answer: 6

Answer

6

The correct answer is

6

. The initial temperature of the sample is

80^\circ\text{C}

. For the first

t

minutes, the temperature decreases by

1.5t^\circ\text{C}

. For the next

12

minutes, the rate is

1.5 + 0.75 = 2.25^\circ\text{C}

per minute, resulting in a temperature decrease of

2.25 \times 12 = 27^\circ\text{C}

. The final temperature equation is

80 - 1.5t - 27 = 44

, which simplifies to

53 - 1.5t = 44

. Solving for

t

gives

1.5t = 9

, or

t = 6

.

Step-by-Step Solution

1

Determine the cooling rate for both phases of the program.

The initial rate is

1.5^\circ\text{C}

per minute. The increased rate is

1.5 + 0.75 = 2.25^\circ\text{C}

per minute.

The rate in the second phase is increased by

0.75^\circ\text{C}

per minute from the initial rate.

2

Set up the linear equation representing the temperature change over time.

80 - 1.5t - 2.25(12) = 44

The final temperature is the initial temperature minus the temperature drops during each phase.

3

Solve the equation to isolate the variable

t

.

80 - 1.5t - 27 = 44 \implies 53 - 1.5t = 44 \implies -1.5t = -9 \implies t = 6

Perform basic algebraic operations to find the value of

t

.

Key Concept

Setting up and solving a linear equation in one variable from a real-world scenario.

Question 277Question →

A community food bank is preparing two types of relief packages: standard boxes and family boxes. Let $x$ represent the number of standard boxes and $y$ represent the number of family boxes. The food bank needs to prepare at least 120 boxes in total. Each standard box contains 3 cans of soup and 2 packages of grain. Each family box contains 6 cans of soup and 5 packages of grain. The food bank has a maximum of 600 cans of soup and a maximum of 450 packages of grain available. Which of the following systems of inequalities represents this situation?

\begin{aligned} x + y &\le 120 \\ 3x + 6y &\ge 600 \\ 2x + 5y &\ge 450 \end{aligned}

\begin{aligned} x + y &\ge 120 \\ 6x + 3y &\le 600 \\ 5x + 2y &\le 450 \end{aligned}

\begin{aligned} x + y &\ge 120 \\ 3x + 6y &\le 600 \\ 2x + 5y &\le 450 \end{aligned}

\begin{aligned} x + y &\ge 120 \\ 3x + 2y &\le 600 \\ 6x + 5y &\le 450 \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} x + y &\ge 120 \\ 3x + 6y &\le 600 \\ 2x + 5y &\le 450 \end{aligned}$

Answer

The system of inequalities with

x + y \ge 120

,

3x + 6y \le 600

, and

2x + 5y \le 450

.

The correct system of inequalities translates the constraint of preparing at least 120 boxes as

x + y \ge 120

. The maximum limit of 600 cans of soup translates as

3x + 6y \le 600

, since each standard box contains 3 cans and each family box contains 6. The maximum limit of 450 packages of grain translates as

2x + 5y \le 450

, since each standard box contains 2 packages and each family box contains 5.

Step-by-Step Solution

1

Translate the total package constraint.

x + y \ge 120

The phrase 'at least 120 boxes in total' indicates that the sum of

x

and

y

must be greater than or equal to 120.

2

Translate the soup constraint.

3x + 6y \le 600

Since each standard box (

x

) has 3 cans of soup and each family box (

y

) has 6 cans of soup, the total soup used is

3x + 6y

. The food bank has 'a maximum of 600 cans,' so this total must be less than or equal to 600.

3

Translate the grain constraint.

2x + 5y \le 450

Since each standard box (

x

) has 2 packages of grain and each family box (

y

) has 5 packages of grain, the total grain used is

2x + 5y

. The food bank has 'a maximum of 450 packages,' so this total must be less than or equal to 450.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 30s

Question 278Question →

In the $xy$ -plane, the system of equations below has infinitely many solutions:

\begin{aligned} \frac{1}{2}(ax - by) &= 7 \\ 3x - 2y &= c \end{aligned}

where $a$ , $b$ , and $c$ are constants. If the line representing the first equation passes through the point $(4, 1)$ , what is the value of $c$ ?

5

8

10

20

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. For a system of linear equations in two variables to have infinitely many solutions, the two equations must describe the exact same line, meaning their coefficients and constant terms are proportional. Multiplying the first equation by

2

yields

ax - by = 14

. Comparing this with

3x - 2y = c

shows that

\frac{a}{3} = \frac{b}{2} = \frac{14}{c}

, which simplifies to

b = \frac{2}{3}a

and

c = \frac{42}{a}

. Since the first line passes through

(4, 1)

, we substitute

x = 4

and

y = 1

to get

4a - b = 14

. Substituting

b = \frac{2}{3}a

gives

4a - \frac{2}{3}a = 14

, which simplifies to

\frac{10}{3}a = 14

, or

a = 4.2

. Finally, solving for

c

gives

c = \frac{42}{4.2} = 10

.

Step-by-Step Solution

1

Clear the fraction in the first equation by multiplying both sides by

2

.

ax - by = 14

This puts the first equation into standard form, making it easier to compare with the second equation.

2

Set up the proportionality of the coefficients for the two equations to represent the same line.

\frac{a}{3} = \frac{-b}{-2} = \frac{14}{c} \implies b = \frac{2}{3}a

and

c = \frac{42}{a}

For a system of two linear equations to have infinitely many solutions, the equations must be equivalent, meaning their corresponding coefficients and constants must be proportional.

3

Substitute the given point

(4, 1)

into the first equation.

a(4) - b(1) = 14 \implies 4a - b = 14

Since the line passes through the point

(4, 1)

, the coordinates must satisfy the equation of the line.

4

Substitute

b = \frac{2}{3}a

into

4a - b = 14

and solve for

a

.

4a - \frac{2}{3}a = 14 \implies \frac{10}{3}a = 14 \implies a = 4.2

This reduces the equation to a single variable, allowing us to find the value of the parameter

a

.

5

Substitute

a = 4.2

into the expression for

c

to find its value.

c = \frac{42}{4.2} = 10

This uses the coefficient proportionality relation from step 2 to determine the constant term of the second equation.

Key Concept

Systems of linear equations with infinitely many solutions require the equations to represent the same line, meaning their coefficients and constant terms are proportional.

Question 279Question →

An industrial oven is used in a bakery. The temperature of the oven chamber, $C$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), $t$ minutes after the heating element is turned on is modeled by the linear equation:

C = 18.5t + 72

After a system upgrade, the starting temperature of the oven is $8^\circ\text{F}$ warmer, and the rate at which the oven heats up is $20\%$ faster. During a test of the upgraded oven, the heating element is turned on for $15$ minutes, after which the oven is turned off and cools down at a constant rate of $12^\circ\text{F}$ per minute. If the cooling process is also linear, which of the following functions models the temperature of the upgraded oven, $U$ , in degrees Fahrenheit, $m$ minutes after it is turned off?

U(m) = -12m + 357.5

U(m) = -12m + 483.9

U(m) = -12m + 413

U(m) = 12m + 413

Show answer & explanation

Answer: $U(m) = -12m + 413$

Answer

The function

U(m) = -12m + 413

models the temperature of the upgraded oven,

U

, in degrees Fahrenheit,

m

minutes after it is turned off.

To find the temperature model during the cooling phase, we must first determine the state of the oven when the cooling begins. The upgraded oven has a starting temperature of

72 + 8 = 80^\circ\text{F}

and a heating rate of

18.5 \times 1.20 = 22.2^\circ\text{F}

per minute. After

15

minutes of heating, the temperature reaches

22.2 \times 15 + 80 = 413^\circ\text{F}

. When the oven is turned off at

m = 0

minutes, its temperature is

413^\circ\text{F}

, which serves as the y-intercept of the cooling function. Since the temperature decreases at a constant rate of

12^\circ\text{F}

per minute, the rate of change (slope) is

-12

. Therefore, the linear model is the function showing a rate of change of

-12

and a starting value of

413

.

Step-by-Step Solution

1

Determine the upgraded starting temperature and heating rate of the oven.

The upgraded starting temperature is

80^\circ\text{F}

and the upgraded heating rate is

22.2^\circ\text{F}

per minute.

The original starting temperature of

72^\circ\text{F}

is increased by

8^\circ\text{F}

to get

72 + 8 = 80^\circ\text{F}

. The original heating rate (slope) of

18.5^\circ\text{F}

per minute is increased by

20\%

, which is calculated as

18.5 \times 1.20 = 22.2^\circ\text{F}

per minute.

2

Calculate the temperature of the upgraded oven at the moment it is turned off.

The temperature is

413^\circ\text{F}

at

t = 15

minutes.

The heating phase is modeled by the linear relationship

H(t) = 22.2t + 80

. Substituting

t = 15

yields

H(15) = 22.2(15) + 80 = 333 + 80 = 413^\circ\text{F}

.

3

Construct the linear function for the cooling phase.

U(m) = -12m + 413

When the oven is turned off (

m = 0

), its temperature is

413^\circ\text{F}

. Since it cools down at a constant rate of

12^\circ\text{F}

per minute, the slope of the cooling function is

-12

. Thus, the linear model is

U(m) = -12m + 413

.

Key Concept

Interpreting and modifying parameters of linear models in multi-stage contextual scenarios.

Alternative Method

Instead of writing the heating function explicitly, you can calculate the total temperature increase directly: the temperature rises by

18.5 \times 1.20 = 22.2^\circ\text{F}

per minute for

15

minutes, which is a total increase of

22.2 \times 15 = 333^\circ\text{F}

. Adding this increase to the upgraded starting temperature of

72 + 8 = 80^\circ\text{F}

gives the peak temperature of

80 + 333 = 413^\circ\text{F}

. Since the cooling phase is linear with a slope of

-12

and a y-intercept of

413

, the function is immediately determined.

Estimated Time:3m 0s

Question 280Question →

To rent an electric scooter, a rider pays a flat unlocking fee plus a fee for each minute of the ride. The total cost $C$ , in dollars, for a ride of $m$ minutes is given by the equation $C = 0.15m + 1.20$ . Which of the following is the best interpretation of the number $1.20$ in this context?

The cost, in dollars, for each minute of the scooter ride

The flat unlocking fee, in dollars, to rent the scooter

The total cost, in dollars, of a scooter ride that lasts 1 minute

The cost, in dollars, for a ride that lasts 15 minutes

Show answer & explanation

Answer: The flat unlocking fee, in dollars, to rent the scooter

Answer

The flat unlocking fee, in dollars, to rent the scooter

In the linear model

C = 0.15m + 1.20

, the constant term

1.20

represents the value of

C

when

m = 0

. Since

m

represents the number of minutes,

m = 0

corresponds to the beginning of the rental. Therefore,

1.20

represents the initial flat unlocking fee, in dollars, to rent the scooter.

Step-by-Step Solution

1

Identify the component of the linear equation

C = 0.15m + 1.20

that corresponds to the number

1.20

.

The number

1.20

is the constant term (y-intercept) of the equation.

A linear equation in slope-intercept form is

y = mx + b

, where

b

is the y-intercept (the value of

y

when

x = 0

).

2

Determine the physical meaning of

m = 0

in the given context.

m = 0

represents a ride that lasts

0

minutes, meaning no time has elapsed yet.

Evaluating the relationship at the initial state helps identify the physical meaning of the y-intercept.

3

Substitute

m = 0

into the equation to find the corresponding cost.

C = 0.15(0) + 1.20 = 1.20

dollars.

This shows that the initial cost, or flat unlocking fee before starting the ride, is

1.20

dollars.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Algebra

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