Algebra Practice Questions — SAT — Page 17

Question 321Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(3, 11)$ and $(7, 23)$ . The function $g$ is defined by $g(x) = f(2x) - 5$ . What is the value of $g(4)$ ?

Show answer & explanation

Answer: 21

Answer

21

The correct answer is 21. First, find the slope of the linear function

f

using the given points

(3, 11)

and

(7, 23)

:

m = \frac{23 - 11}{7 - 3} = 3

. The equation of the line is

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

. To find

g(4)

, substitute

x = 4

into the definition of

g(x)

:

g(4) = f(2 \cdot 4) - 5 = f(8) - 5

. Evaluating

f(8)

gives

3(8) + 2 = 26

. Finally, subtracting

5

gives

g(4) = 26 - 5 = 21

.

Step-by-Step Solution

1

Determine the equation of the linear function

f(x)

f(x) = 3x + 2

First find the slope

m = \frac{23 - 11}{7 - 3} = 3

. Then, use the point-slope formula with

(3, 11)

to get

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

.

2

Express

g(4)

in terms of

f

g(4) = f(8) - 5

Substitute

x = 4

into the definition

g(x) = f(2x) - 5

to get

g(4) = f(2(4)) - 5

.

3

Calculate the value of

f(8)

and

g(4)

g(4) = 21

Evaluate

f(8) = 3(8) + 2 = 26

, then subtract

5

to obtain

g(4) = 26 - 5 = 21

.

Key Concept

Linear Functions and Graphs

Question 322Question →

A scientist is monitoring the population of two species of bacteria, Species A and Species B, in a controlled environment. Initially, the combined population of the two species is $12,000$ . Over the next week, the population of Species A doubles, and the population of Species B triples. If the total combined population of both species is $29,000$ at the end of the week, what was the initial population of Species A?

Show answer & explanation

Answer: 7000

Answer

The initial population of Species A was 7000.

To find the initial population of Species A, we set up a system of linear equations representing the total initial population and the population after one week. Let

x

represent the initial population of Species A and

y

represent the initial population of Species B. The initial combined population is given by

x + y = 12,000

. After one week, the population of Species A doubles (

2x

) and Species B triples (

3y

), so the new combined population is

2x + 3y = 29,000

. We can express

y

in terms of

x

as

y = 12,000 - x

. Substituting this expression into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing and simplifying yields

-x + 36,000 = 29,000

, which simplifies to

x = 7,000

. Therefore, the initial population of Species A was 7,000.

Step-by-Step Solution

1

Define the variables and set up the first equation.

x + y = 12,000

Let

x

be the initial population of Species A and

y

be the initial population of Species B. Their combined initial population is

12,000

.

2

Set up the second equation based on the growth after one week.

2x + 3y = 29,000

The population of Species A doubles to

2x

and Species B triples to

3y

, summing to a total of

29,000

.

3

Solve for

x

using substitution.

x = 7,000

From the first equation,

y = 12,000 - x

. Substituting this into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing the

3

yields

2x + 36,000 - 3x = 29,000

. Combining like terms results in

-x + 36,000 = 29,000

. Subtracting

36,000

from both sides gives

-x = -7,000

, which simplifies to

x = 7,000

.

Key Concept

Solving systems of linear equations using substitution or elimination methods

Question 323Question →

An artist creates $x$ small sculptures and $y$ large sculptures. Each small sculpture requires 3 hours of crafting and 2 hours of painting. Each large sculpture requires 7 hours of crafting and 3 hours of painting. The artist can spend at most 120 hours on crafting and at most 50 hours on painting. Which of the following pairs of small and large sculptures can the artist create under these constraints?

5 small sculptures and 15 large sculptures

12 small sculptures and 9 large sculptures

10 small sculptures and 8 large sculptures

15 small sculptures and 12 large sculptures

Show answer & explanation

Answer: 10 small sculptures and 8 large sculptures

Answer

10 small sculptures and 8 large sculptures

The option featuring 10 small sculptures and 8 large sculptures is correct because substituting

x = 10

and

y = 8

satisfies both linear inequalities representing the constraints. Specifically, the crafting time of 86 hours is less than or equal to the maximum allowed 120 hours (

3(10) + 7(8) = 86 \leq 120

), and the painting time of 44 hours is less than or equal to the maximum allowed 50 hours (

2(10) + 3(8) = 44 \leq 50

).

Step-by-Step Solution

1

Set up the system of inequalities representing the constraints.

The crafting constraint is

3x + 7y \leq 120

. The painting constraint is

2x + 3y \leq 50

. The variables must be non-negative:

x \geq 0

and

y \geq 0

.

To model the limits on crafting hours and painting hours mathematically.

2

Substitute the values of each option into the system to verify which one satisfies both inequalities.

For the option with 10 small sculptures and 8 large sculptures (

x = 10, y = 8

):

3(10) + 7(8) = 30 + 56 = 86 \leq 120

(True)

2(10) + 3(8) = 20 + 24 = 44 \leq 50

(True)

Only a coordinate pair that makes both inequality statements true is a valid solution.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 324Question →

During a dry season, the volume of water $V$ , in millions of gallons, in a municipal reservoir $d$ days after the start of a monitoring period is modeled by the equation $V = 248 - 1.8d$ , where $0 \leq d \leq 60$ . According to the model, at what rate, in gallons per minute, is water leaving the reservoir?

Show answer & explanation

Answer: 1250

Answer

The reservoir is losing water at a rate of

1,250

gallons per minute.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, indicating a decrease of

1.8

million gallons of water per day. To express this rate in gallons per minute, we perform the unit conversion. First, convert

1.8

million gallons to gallons:

1.8 \times 1,000,000 = 1,800,000

gallons. Second, convert

1

day to minutes:

24 \text{ hours/day} \times 60 \text{ minutes/hour} = 1,440

minutes. Finally, divide the total gallons by the total minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

gallons per minute.

Step-by-Step Solution

1

Identify the daily rate of water leaving the reservoir from the given linear equation.

The daily rate is

1.8

million gallons per day.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, which represents the rate of change of the volume with respect to time in days. The negative sign indicates a decrease, meaning water is leaving at a rate of

1.8

million gallons per day.

2

Convert the volume from million gallons to gallons.

1,800,000

gallons.

To convert from millions of gallons to gallons, multiply the value by

1,000,000

, resulting in

1.8 \times 1,000,000 = 1,800,000

.

3

Convert the time unit from days to minutes.

1,440

minutes.

There are

24

hours in a day and

60

minutes in an hour, so

1 \text{ day} = 24 \times 60 = 1,440 \text{ minutes}

.

4

Calculate the rate in gallons per minute.

1,250

gallons per minute.

Divide the volume in gallons by the time in minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

.

Key Concept

Interpreting the slope of a linear relationship in context and performing unit conversions on rates.

Question 325Question →

A tour guide offers two types of guided tours: a walking tour and a trolley tour. On Saturday, the guide led $3$ walking tours and $2$ trolley tours, which lasted a total of $8$ hours. On Sunday, the guide led $2$ walking tours and $4$ trolley tours, which lasted a total of $12$ hours. If each walking tour has a duration of $w$ hours and each trolley tour has a duration of $t$ hours, what is the total duration, in hours, of $1$ walking tour and $1$ trolley tour?

2.0

3.5

4.0

4.25

Show answer & explanation

Answer: 3.5

Answer

The correct answer is 3.5.

The correct answer is

3.5

. The scenario can be represented by the system of linear equations

3w + 2t = 8

and

2w + 4t = 12

. Multiplying the first equation by

2

gives

6w + 4t = 16

. Subtracting the second equation from this yields

4w = 4

, which simplifies to

w = 1

. Substituting

w = 1

into the first equation yields

3(1) + 2t = 8

, which simplifies to

2t = 5

or

t = 2.5

. The sum of the durations of one walking tour and one trolley tour is

w + t = 1 + 2.5 = 3.5

hours.

Step-by-Step Solution

1

Represent the given scenarios with a system of linear equations using the variables

w

(duration of a walking tour) and

t

(duration of a trolley tour).

The system of equations is:
1)

3w + 2t = 8

2)

2w + 4t = 12

This translates the word problem into mathematical equations.

2

Solve the system of equations for

w

and

t

. For example, multiply the first equation by

2

to align the

t

coefficients, and then subtract the second equation from it.

Multiplying the first equation by

2

gives

6w + 4t = 16

.
Subtracting the second equation (

2w + 4t = 12

) from this yields

4w = 4

, which simplifies to

w = 1

.

This eliminates the variable

t

to find the value of

w

.

3

Substitute

w = 1

back into one of the original equations to solve for

t

.

Using the first equation:

3(1) + 2t = 8 \implies 3 + 2t = 8 \implies 2t = 5 \implies t = 2.5

.

This determines the value of the second variable,

t

.

4

Calculate the sum of

w

and

t

to find the total duration of

1

walking tour and

1

trolley tour.

w + t = 1 + 2.5 = 3.5

.

This directly answers the question asked in the problem stem.

Key Concept

Solving a system of linear equations to find a linear combination of variables

Estimated Time:1m 30s

Question 326Question →

A solar power system stores electricity in a battery at a constant rate during the day. The total amount of energy in the battery, in kilowatt-hours (kWh), is a linear function of the number of hours after sunrise. Exactly $3$ hours after sunrise, the battery contains $14$ kWh of energy, and exactly $7$ hours after sunrise, it contains $26$ kWh of energy. If the battery's maximum capacity is $35$ kWh, how many hours after sunrise will it take for the battery to be fully charged?

Show answer & explanation

Answer: 10

Answer

10

To find the number of hours after sunrise it takes to fully charge the battery to 35 kWh, we first determine the constant rate of energy storage. The rate of change is given by the difference in energy divided by the difference in hours:

\frac{26 - 14}{7 - 3} = \frac{12}{4} = 3

kWh per hour. The relationship between energy

E

and hours

h

can be written as

E = 3h + b

. Substituting the known point

(3, 14)

gives

14 = 3(3) + b

, which simplifies to

b = 5

. Thus, the linear equation representing this relationship is

E = 3h + 5

. To find the time when the battery is fully charged to

35

kWh, set

E = 35

in the equation:

35 = 3h + 5

. Subtracting 5 from both sides yields

30 = 3h

, and dividing by 3 gives

h = 10

hours.

Step-by-Step Solution

1

Calculate the rate of energy storage per hour.

3 kWh per hour

Since the energy increases at a constant rate, the rate is the change in energy divided by the change in time:

\frac{26 - 14}{7 - 3} = 3

.

2

Determine the initial energy stored at sunrise.

5 kWh

Using the rate of 3 kWh per hour and the fact that the battery had 14 kWh at 3 hours, we subtract the energy gained in those 3 hours:

14 - 3(3) = 5

.

3

Set up the linear equation and solve for the time when the battery reaches 35 kWh.

10 hours

The linear relationship is

E = 3h + 5

. Setting

E = 35

gives

35 = 3h + 5

. Subtracting 5 from both sides gives

30 = 3h

, which simplifies to

h = 10

.

Key Concept

Linear Equations in Two Variables

Question 327Question →

In the $xy$ -plane, the graph of the linear equation $3x + by = 36$ , where $b$ is a constant, is a line with a slope of $-0.75$ . What is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The value of

b

is

4

.

The linear equation

3x + by = 36

can be rewritten in slope-intercept form by subtracting

3x

from both sides to get

by = -3x + 36

, and then dividing all terms by

b

to get

y = -\frac{3}{b}x + \frac{36}{b}

. The slope of this line is the coefficient of

x

, which is

-\frac{3}{b}

. Setting this equal to the given slope of

-0.75

(or

-\frac{3}{4}

) gives

-\frac{3}{b} = -\frac{3}{4}

. Solving for

b

yields

b = 4

.

Step-by-Step Solution

1

Express the given linear equation in slope-intercept form.

y = -\frac{3}{b}x + \frac{36}{b}

To identify the slope of the line in terms of the constant

b

, we rewrite the equation

3x + by = 36

in the form

y = mx + d

.

2

Equate the expression for the slope to the given slope value.

-\frac{3}{b} = -0.75

The coefficient of

x

in the slope-intercept form represents the slope of the line, which is given as

-0.75

.

3

Solve the equation for

b

.

b = 4

Multiply both sides of the equation by

-1

to get

\frac{3}{b} = 0.75

. Since

0.75 = \frac{3}{4}

, we have

\frac{3}{b} = \frac{3}{4}

, which gives

b = 4

.

Key Concept

Converting a linear equation from standard form to slope-intercept form to determine its slope.

Estimated Time:1m 30s

Question 328Question →

A researcher is monitoring the temperature, $T$ , in degrees Celsius, of a chemical reaction. The target temperature range is maintained by an automated cooling system that activates when the temperature satisfies the inequality $-5(T - 12) - 8(3 - 2T) < 14(T + 3)$ . Which of the following inequalities represents all possible values of $T$ for which the cooling system will activate?

T < -2

T > -2

T > -42

T > 2

Show answer & explanation

Answer: $T > -2$

Answer

The inequality representing all possible values of the temperature is

T > -2

.

The correct inequality is obtained by first expanding both sides of the inequality to get

-5T + 60 - 24 + 16T < 14T + 42

. Simplifying the left side yields

11T + 36 < 14T + 42

. Subtracting

14T

from both sides gives

-3T + 36 < 42

, and subtracting

36

from both sides gives

-3T < 6

. Finally, dividing both sides by

-3

and reversing the inequality sign results in

T > -2

.

Step-by-Step Solution

1

Distribute the constants on both sides of the inequality:

-5(T - 12) - 8(3 - 2T) < 14(T + 3)

.

-5T + 60 - 24 + 16T < 14T + 42

Remove the parentheses to prepare the inequality for simplification.

2

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

11T + 36 < 14T + 42

Simplify the expression before isolating the variable.

3

Subtract

14T

and

36

from both sides of the inequality.

-3T < 6

Group all variable terms on one side and constant terms on the other.

4

Divide both sides of the inequality by

-3

and reverse the inequality sign.

T > -2

Dividing by a negative number reverses the inequality relationship.

Key Concept

Solving multi-step linear inequalities in one variable, distributing coefficients, combining like terms, and reversing the inequality sign when dividing by a negative number.

Estimated Time:2m 0s

Question 329Question →

A commercial 3D printer uses a polymer filament to print prototype parts. The remaining mass of the filament spool, $M$ , in grams, is modeled by a linear function of the total printing time, $t$ , in minutes. When the printer has been running for $20$ minutes, the spool has $850$ grams of filament remaining. When the printer has been running for $50$ minutes, the spool has $685$ grams of filament remaining. According to the model, how many grams of filament does the printer use per minute?

Show answer & explanation

Answer: 5.5

Answer

The printer uses 5.5 grams of filament per minute.

The relationship between the printing time

t

and the remaining mass of the filament

M

is linear. The slope of this line represents the change in filament mass per unit of time. Based on the given coordinates

(20, 850)

and

(50, 685)

, the slope is calculated as

\frac{685 - 850}{50 - 20} = \frac{-165}{30} = -5.5

grams per minute. This means that the remaining mass decreases by

5.5

grams for each additional minute the printer runs, which corresponds to a usage rate of

5.5

grams of filament per minute.

Step-by-Step Solution

1

Identify the data points representing time and remaining filament mass from the context.

The two points are

(20, 850)

and

(50, 685)

.

These points allow us to calculate the rate at which the mass of the filament is changing over time.

2

Calculate the change in remaining mass and the change in printing time.

Change in mass is

685 - 850 = -165

grams. Change in time is

50 - 20 = 30

minutes.

Finding the differences in the dependent variable (mass) and independent variable (time) is the standard method to calculate a rate of change.

3

Calculate the rate of filament usage per minute.

\frac{-165\text{ grams}}{30\text{ minutes}} = -5.5

grams per minute. The rate of usage is the positive magnitude, which is

5.5

grams per minute.

The slope of the linear relationship is negative because the mass is decreasing, but the rate of consumption/usage is represented as a positive quantity.

Key Concept

Interpreting the slope of a linear function in context as a rate of change.

Question 330Question →

An agricultural drone is used to spray fertilizer on a crop field. The volume of fertilizer remaining in the drone's tank, $F$ , in liters, after the drone has been spraying for $t$ minutes is modeled by the equation $F = 240 - 3.2t$ , where $0 \leq t \leq 75$ . What is the best interpretation of the number $3.2$ in this context?

The initial volume of fertilizer, in liters, in the tank before spraying begins.

The number of minutes required for the drone to spray

1

liter of fertilizer.

The volume of fertilizer, in liters, sprayed from the tank each minute.

The total number of minutes the drone can spray before the tank is completely empty.

Show answer & explanation

Answer: The volume of fertilizer, in liters, sprayed from the tank each minute.

Answer

The volume of fertilizer, in liters, sprayed from the tank each minute.

In the linear relationship

F = 240 - 3.2t

, the variable

t

represents time in minutes, and

F

represents the remaining volume in liters. The coefficient of

t

, which is

-3.2

, represents the rate of change of the volume of fertilizer in the tank. A slope of

-3.2

means that the volume decreases by

3.2

liters for every additional minute of spraying. Therefore, the number

3.2

represents the volume of fertilizer, in liters, that is sprayed from the tank each minute.

Step-by-Step Solution

1

Identify the structure of the linear equation.

The equation is of the form

y = mx + b

, or in this case,

F = -3.2t + 240

, where the slope is

-3.2

and the

F

-intercept (initial value) is

240

.

Analyzing the components of a linear function helps determine what each constant represents in context.

2

Determine the meaning of the slope in context.

The slope is the coefficient of the independent variable

t

(time in minutes). The value

-3.2

indicates that for each increase of

1

minute in spraying time, the remaining volume of fertilizer,

F

, decreases by

3.2

liters.

Slope represents the rate of change of the dependent variable relative to the independent variable.

3

Interpret the absolute value of the slope.

The value

3.2

represents the magnitude of this rate, which is the amount of fertilizer sprayed out of the tank per minute (

3.2

liters per minute).

Matching the mathematical rate of change to the corresponding verbal description provides the correct interpretation.

Key Concept

Slope in a linear model represents the constant rate of change of the dependent variable per unit increase of the independent variable.

Estimated Time:1m 15s

Question 331Question →

A commercial drone delivery service models the total power remaining in a drone's battery, $P(w)$ , as a percentage, after carrying a payload of weight $w$ kilograms for a fixed delivery distance of 10 kilometers. The relationship is modeled by the linear function $P(w) = -8.5w + 92$ . Which of the following is the best interpretation of the value 92 in this context?

The decrease in the remaining battery power, as a percentage, for each increase of 1 kilogram in payload weight.

The remaining battery power, as a percentage, after the drone carries a payload of 1 kilogram for the delivery distance.

The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

The payload weight, in kilograms, for which the remaining battery power after the delivery is 0 percent.

Show answer & explanation

Answer: The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

Answer

The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

The linear equation is written in the slope-intercept form

P(w) = mw + b

, where

b = 92

is the vertical intercept (y-intercept). This intercept represents the value of the dependent variable,

P(w)

, when the independent variable,

w

, is equal to

0

. In this context,

w

is the payload weight in kilograms and

P(w)

is the remaining battery power as a percentage. Thus, the value

92

represents the remaining battery power, as a percentage, when the drone carries a payload of

0

kilograms.

Step-by-Step Solution

1

Identify the structure of the linear function and the component being interpreted.

The linear equation is given in slope-intercept form,

P(w) = mw + b

, where

m = -8.5

is the slope and

b = 92

is the

P(w)

-intercept (y-intercept).

Understanding the components of a linear function allows us to relate them to their contextual definitions.

2

Evaluate the value of the function at the

P(w)

-intercept where the independent variable is zero.

Substitute

w = 0

into the equation:

P(0) = -8.5(0) + 92 = 92

.

The y-intercept represents the value of the dependent variable when the independent variable is equal to 0.

3

Interpret the meaning of

w = 0

and

P(0) = 92

in the given context.

Since

w

represents the payload weight in kilograms and

P(w)

represents the remaining battery power as a percentage,

w = 0

corresponds to a payload of 0 kilograms, and

P(0) = 92

corresponds to a remaining battery power of 92%.

Applying the units and definitions of the variables completes the contextual interpretation of the constant.

Key Concept

Interpreting the y-intercept of a linear function in a real-world context

Question 332Question →

In the $xy$ -plane, the graph of the linear equation $y - 4 = m(x + 2)$ contains the point $(6, 8)$ , where $m$ is a constant. What is the value of the $x$ -intercept of the line?

-10

-6

-4

5

Show answer & explanation

Answer: -10

Answer

-10

The correct answer is

-10

. By substituting the point

(6, 8)

into the equation

y - 4 = m(x + 2)

, we obtain

8 - 4 = m(6 + 2)

, which simplifies to

4 = 8m

, giving

m = 0.5

. Using this value of

m

, the equation of the line is

y - 4 = 0.5(x + 2)

. The

x

-intercept is the value of

x

when

y = 0

. Substituting

0

for

y

gives

-4 = 0.5(x + 2)

, which simplifies to

-8 = x + 2

. Solving for

x

gives

x = -10

.

Step-by-Step Solution

1

Substitute the given point

(6, 8)

into the equation to find the value of the constant

m

.

8 - 4 = m(6 + 2) \Rightarrow 4 = 8m \Rightarrow m = 0.5

Since the line passes through the point

(6, 8)

, these coordinates must satisfy the equation.

2

Write the completed equation of the line using the value of

m

.

y - 4 = 0.5(x + 2)

Substituting

m = 0.5

back into the original point-slope equation gives the specific line.

3

Find the

x

-intercept by setting

y = 0

and solving for

x

.

0 - 4 = 0.5(x + 2) \Rightarrow -4 = 0.5x + 1 \Rightarrow -5 = 0.5x \Rightarrow x = -10

The

x

-intercept of a graph is the point where the graph crosses the

x

-axis, which occurs where

y = 0

.

Key Concept

Finding the intercepts and slope of a linear equation in two variables

Question 333Question →

A scientist is measuring the temperature of a chemical compound during an experiment. The temperature of the compound decreases at a constant rate. At $t = 4$ minutes after the experiment begins, the temperature of the compound is $72^\circ\text{C}$ . At $t = 12$ minutes, the temperature of the compound is $48^\circ\text{C}$ . Which of the following functions $T$ models the temperature, in degrees Celsius, of the compound $t$ minutes after the experiment begins, where $0 \leq t \leq 25$ ?

T(t) = -3t + 84

T(t) = -3t + 72

T(t) = -\frac{1}{3}t + \frac{220}{3}

T(t) = -3t + 60

Show answer & explanation

Answer: $T(t) = -3t + 84$

Answer

T(t) = -3t + 84

To find the linear model

T(t) = mt + b

, we first calculate the slope

m

using the given points

(4, 72)

and

(12, 48)

. The slope is the change in temperature divided by the change in time:

m = \frac{48 - 72}{12 - 4} = -3

. Next, we find the y-intercept

b

by substituting one of the points into the equation

T(t) = mt + b

. Substituting

t = 4

and

T(4) = 72

yields

72 = -3(4) + b

, which simplifies to

72 = -12 + b

. Adding

12

to both sides gives

b = 84

. Therefore, the correct linear function modeling the temperature is

T(t) = -3t + 84

.

Step-by-Step Solution

1

Calculate the rate of temperature change (slope) per minute.

The slope

m

is

-3

.

Since the temperature decreases at a constant rate, the relationship is linear. The slope

m

is the change in temperature divided by the change in time:

m = \frac{48 - 72}{12 - 4} = \frac{-24}{8} = -3

.

2

Find the initial temperature (y-intercept) at

t = 0

.

The y-intercept

b

is

84

.

Substitute the point

(4, 72)

and the slope

m = -3

into the slope-intercept equation

T(t) = mt + b

. This gives

72 = -3(4) + b

, which simplifies to

72 = -12 + b

. Adding

12

to both sides yields

b = 84

.

3

Write the final equation.

T(t) = -3t + 84

Combine the slope of

-3

and the y-intercept of

84

to complete the linear model equation.

Key Concept

Writing and interpreting linear functions from two points

Alternative Method

Instead of deriving the function algebraically, substitute the given values of

t = 4

and

t = 12

into each of the given options. Only the correct option yields the corresponding temperatures of 72 and 48 degrees Celsius, respectively.

Estimated Time:1m 30s

Question 334Question →

Consider the system of linear equations below:

\begin{aligned} 3x + 4y &= 8 \\ ax + 8y &= 15 \end{aligned}

If the system of equations has no solution, and $a$ is a constant, what is the value of $a$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

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

3x + 4y = 8

in slope-intercept form gives

y = -\frac{3}{4}x + 2

, so its slope is

-\frac{3}{4}

. Writing

ax + 8y = 15

in slope-intercept form gives

y = -\frac{a}{8}x + \frac{15}{8}

, so its slope is

-\frac{a}{8}

. Equating the slopes gives

-\frac{3}{4} = -\frac{a}{8}

. Solving for

a

gives

a = 6

. Since the y-intercepts (

2

and

\frac{15}{8}

) are different, the lines are parallel and have no intersection points.

Step-by-Step Solution

1

Convert the first equation

3x + 4y = 8

to slope-intercept form.

y = -\frac{3}{4}x + 2

To find the slope of the first line, which is

-\frac{3}{4}

.

2

Convert the second equation

ax + 8y = 15

to slope-intercept form.

y = -\frac{a}{8}x + \frac{15}{8}

To find the slope of the second line in terms of

a

, which is

-\frac{a}{8}

.

3

Equate the slopes of the two lines.

-\frac{3}{4} = -\frac{a}{8}

Parallel lines have the same slope, and a system with parallel lines has no solution.

4

Solve the equation for

a

.

a = 6

Isolating the variable

a

yields the value of the constant.

Key Concept

A system of linear equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Question 335Question →

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

$x$	$f(x)$
$2$	$5$
$5$	$14$
$8$	$23$

If the function $g$ is defined by $g(x) = f(x) + k$ , where $k$ is a constant, and the $x$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane is $-2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

To find the value of

k

, we first determine the linear function

f(x)

using the values in the table. The slope of

f

is calculated as

\frac{14 - 5}{5 - 2} = 3

. Substituting the point

(2, 5)

into the slope-intercept form

f(x) = 3x + b

gives

5 = 3(2) + b

, which simplifies to

b = -1

. Therefore,

f(x) = 3x - 1

. The function

g

is defined as

g(x) = f(x) + k = 3x - 1 + k

. Since the

x

-intercept of the graph of

y = g(x)

is

-2

, we must have

g(-2) = 0

. Substituting

x = -2

into the equation for

g(x)

gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

. Solving for

k

yields

k = 7

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

.

The slope of

f

is

3

.

Using the points

(2, 5)

and

(5, 14)

from the table, the slope is calculated as

m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3

.

2

Find the equation for

f(x)

.

f(x) = 3x - 1

Using the slope-intercept form

f(x) = 3x + b

and the point

(2, 5)

, we solve for

b

:

5 = 3(2) + b

, which gives

b = -1

.

3

Set up the equation for

g(x)

and use the

x

-intercept to find the value of

k

.

k = 7

Since

g(x) = f(x) + k

, we have

g(x) = 3x - 1 + k

. The

x

-intercept of

g(x)

is

-2

, so

g(-2) = 0

. This gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

, meaning

k = 7

.

Key Concept

Finding the equation of a linear function from tabular data and applying translations and intercepts.

Question 336Question →

The temperature, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), of a laboratory incubator $m$ minutes after a cooling cycle begins is modeled by the equation $T = 95 - 0.8(2m - 5)$ . For a specific experiment, the incubator temperature must be at most $75^\circ\text{F}$ . Which of the following inequalities represents all possible values of $m$ for which the incubator temperature meets this requirement?

m \geq 15

m \leq 15

m \geq 10

m \geq 30

Show answer & explanation

Answer: $m \geq 15$

Answer

The correct inequality is

m \geq 15

.

The correct inequality is

m \geq 15

. To find this, set the temperature model

95 - 0.8(2m - 5)

to be less than or equal to

75

. Subtracting

95

from both sides gives

-0.8(2m - 5) \leq -20

. Dividing both sides by

-0.8

requires flipping the inequality sign, which yields

2m - 5 \geq 25

. Adding

5

to both sides gives

2m \geq 30

, and dividing by

2

yields

m \geq 15

.

Step-by-Step Solution

1

Set up the inequality based on the requirement that the temperature must be at most

75^\circ\text{F}

.

95 - 0.8(2m - 5) \leq 75

The phrase 'at most' corresponds to a less-than-or-equal-to sign.

2

Subtract

95

from both sides of the inequality.

-0.8(2m - 5) \leq -20

This begins the process of isolating the term containing the variable

m

.

3

Divide both sides by

-0.8

and flip the inequality sign.

2m - 5 \geq 25

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

4

Add

5

to both sides of the inequality.

2m \geq 30

This isolates the variable term

2m

.

5

Divide both sides by

2

to solve for

m

.

m \geq 15

This yields the solution set for the variable

m

.

Key Concept

Solving multi-step linear inequalities in one variable, specifically expanding using the distributive property and reversing the inequality sign when multiplying or dividing by a negative number.

Question 337Question →

A researcher monitors the temperature of a chemical compound during an experiment. The temperature, $T$ , in degrees Celsius, of the compound $m$ minutes after heating begins can be modeled by a linear relationship. The temperature increases by $4.5$ degrees Celsius every $2$ minutes. If the temperature of the compound was $-12$ degrees Celsius when heating began, after how many minutes will the temperature of the compound reach $15$ degrees Celsius?

Show answer & explanation

Answer: 12

Answer

The temperature of the compound will reach

15

degrees Celsius after

12

minutes.

The correct answer is

12

because the temperature increases at a constant rate of

2.25

degrees Celsius per minute. Starting from an initial temperature of

-12

degrees Celsius, the equation is

T = 2.25m - 12

. Setting

T = 15

gives

15 = 2.25m - 12

, which simplifies to

27 = 2.25m

, resulting in

m = 12

.

Step-by-Step Solution

1

Calculate the constant rate of temperature increase per minute.

2.25

degrees Celsius per minute

This rate represents the slope of the linear relationship between time and temperature.

2

Formulate the linear equation representing the temperature

T

as a function of the elapsed minutes

m

.

T = 2.25m - 12

The slope is

2.25

and the vertical intercept is the initial temperature of

-12

degrees Celsius.

3

Set

T = 15

and solve the linear equation for

m

.

m = 12

This determines the exact number of minutes needed for the temperature to reach

15

degrees Celsius.

Key Concept

Interpreting and solving linear equations modeled from context

Question 338Question →

To restore a depleted wetland, conservationists pump water into a basin. The volume of water in the basin, $W$ , in thousands of gallons, $t$ hours after the pumping begins is modeled by the equation $W = 12.5t + 85$ . According to the model, how many hours of pumping are required for the volume of water in the basin to increase by $150$ thousand gallons?

Show answer & explanation

Answer: 12

Answer

12

The linear equation is given in slope-intercept form,

W = mt + b

, where

m = 12.5

is the slope and

b = 85

is the y-intercept. In this context, the slope

m = 12.5

represents the rate at which water is pumped into the basin, which is

12.5

thousand gallons per hour. To find the number of hours required for the volume of water to increase by

150

thousand gallons, divide the total increase by the rate:

\frac{150}{12.5} = 12

hours.

Step-by-Step Solution

1

Identify the rate of water volume increase per hour from the equation.

The rate is

12.5

thousand gallons per hour.

In the linear equation

W = 12.5t + 85

, the coefficient of

t

(the slope) represents the rate of change of the water volume with respect to time.

2

Set up a relation to find the time

t

for a volume increase of

150

thousand gallons.

12.5t = 150

The change in volume is equal to the rate of change multiplied by the time elapsed.

3

Solve the equation for

t

.

t = 12

Dividing both sides of the equation

12.5t = 150

by

12.5

isolates

t

.

Key Concept

Interpreting the slope of a linear equation in context as a rate of change.

Estimated Time:1m 30s

Question 339Question →

A community art studio offers clay sculpting classes. The studio charges a one-time registration fee plus a fee for each pound of clay used. A student who uses $12$ pounds of clay is charged a total of $\$106$ . A student who uses $25$ pounds of clay is charged a total of $\$197$ . If the relationship between the total charge, $C$ , in dollars, and the amount of clay used, $p$ , in pounds, is linear, which of the following equations models this relationship?

C = 22p + 7

C = 7p + 22

C = 7p + 12

C = \frac{1}{7}p + 22

Show answer & explanation

Answer: $C = 7p + 22$

Answer

C = 7p + 22

The correct equation is

C = 7p + 22

. To find this, we first determine the rate of change, which is the cost per pound of clay. This is calculated as the change in total cost divided by the change in the amount of clay used:

\frac{197 - 106}{25 - 12} = \frac{91}{13} = 7

dollars per pound. Next, we use the linear equation form

C = mp + b

, where

m

is the slope (

7

) and

b

is the y-intercept (the registration fee). Substituting

C = 106

and

p = 12

gives

106 = 7(12) + b

, which simplifies to

106 = 84 + b

. Solving for

b

yields

b = 22

. Therefore, the linear relationship is represented by

C = 7p + 22

.

Step-by-Step Solution

1

Calculate the slope (

m

), which represents the price per pound of clay, using the coordinate points

(12, 106)

and

(25, 197)

.

m = \frac{197 - 106}{25 - 12} = \frac{91}{13} = 7

The slope of a linear equation represents the constant rate of change between the independent variable (pounds of clay) and the dependent variable (total cost).

2

Substitute the slope

m = 7

and one of the points, such as

(12, 106)

, into the slope-intercept form

C = mp + b

to solve for the y-intercept

b

.

106 = 7(12) + b \implies 106 = 84 + b \implies b = 22

The y-intercept represents the flat registration fee, which is the cost when

0

pounds of clay are used.

3

Write the final equation by substituting the calculated slope and y-intercept back into the slope-intercept form.

C = 7p + 22

Combining the constant rate of change and the initial value yields the linear model.

Key Concept

Linear Equations in Two Variables

Question 340Question →

A municipal water treatment facility utilizes 3 identical backup filtration units to process stormwater runoff. The average volume of water, $V(t)$ , in thousands of gallons, remaining to be filtered per unit $t$ hours after the units are activated is modeled by the equation:

3V(t) + 14.4t = 162

where $0 \le t \le 10$ . Which of the following is the best interpretation of the number $14.4$ in this context?

The initial total volume of water, in thousands of gallons, remaining to be filtered by all 3 units combined.

The rate, in thousands of gallons per hour, at which water is filtered by each individual unit, which is 4.8.

The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The rate, in thousands of gallons per hour, at which water is filtered by each individual unit, which is 4.6.

Show answer & explanation

Answer: The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

Answer

The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The correct answer is the option stating that 14.4 represents the rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined. In the given equation

3V(t) + 14.4t = 162

, we can isolate the total volume remaining,

3V(t)

, as

3V(t) = 162 - 14.4t

. In this form, the rate of change of the total volume is

-14.4

, indicating that the combined remaining volume decreases by 14.4 thousand gallons each hour.

Step-by-Step Solution

1

Express the total volume of water remaining to be filtered in terms of the variable

V(t)

.

Since

V(t)

is the average volume of water remaining per unit and there are 3 identical units, the total volume of water remaining to be filtered by all 3 units combined is

3V(t)

.

This allows us to analyze the total filtration system rather than the average per unit.

2

Isolate the term representing the total volume remaining,

3V(t)

, in the given equation.

3V(t) = 162 - 14.4t

This puts the equation into a standard linear form

y = mx + b

where

y = 3V(t)

is the dependent variable (total volume remaining) and

x = t

is the independent variable (time in hours).

3

Interpret the slope and the y-intercept of the isolated linear equation.

The constant term 162 is the initial total volume of water (in thousands of gallons) remaining at

t = 0

. The coefficient of

t

, which is

-14.4

, represents the change in the total volume remaining per hour.

In a linear model

y = mx + b

, the slope

m

represents the rate of change of the dependent variable per unit increase of the independent variable.

4

Determine the contextual meaning of the absolute value of the rate of change, 14.4.

A rate of change of

-14.4

thousand gallons per hour means that the total volume of water remaining decreases by 14.4 thousand gallons each hour. Therefore, 14.4 is the rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The rate at which the remaining volume decreases is equal to the rate at which the system filters the water.

Key Concept

Interpreting coefficients and constants in a linear relationship within a real-world context, particularly when variables represent averages or totals.

Algebra

Answer

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