Algebra Practice Questions — SAT — Page 19

Question 361Question →

A line in the $xy$ -plane representing the linear function $f$ has a $y$ -intercept of $(0, 8)$ and contains the point $(a, b)$ , where $a$ and $b$ are positive integers. If the slope of this line is $-\frac{3}{5}$ and the sum of $a$ and $b$ is $12$ , what is the value of $a$ ?

2

6

8

10

Show answer & explanation

Answer: 10

Answer

10

Since the line representing the function has a y-intercept of (0, 8) and a slope of -3/5, its equation in slope-intercept form is y = -3/5x + 8. Since the line contains the point (a, b), substituting x = a and y = b into this equation gives b = -3/5a + 8. The sum of a and b is given as 12, so a + b = 12, which can be rewritten as b = 12 - a. Substituting this expression for b gives the equation 12 - a = -3/5a + 8. Subtracting 8 from both sides yields 4 - a = -3/5a. Adding a to both sides yields 4 = 2/5a. Multiplying both sides by 5/2 gives a = 10.

Step-by-Step Solution

1

Write the equation of the linear function in slope-intercept form.

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

The y-intercept of the line is given as (0, 8), and the slope of the line is given as -3/5.

2

Substitute the point (a, b) into the equation of the line.

b = -\frac{3}{5}a + 8

Since the point (a, b) lies on the line, its coordinates must satisfy the equation of the line.

3

Express b in terms of a using the given sum relationship.

b = 12 - a

We are given that the sum of a and b is 12, which can be written as a + b = 12.

4

Substitute the expression for b into the line's equation and solve for a.

a = 10

Substituting b = 12 - a into b = -3/5a + 8 yields 12 - a = -3/5a + 8, which simplifies to 4 = 2/5a, leading to a = 10.

Key Concept

Linear Functions and Graphs

Question 362Question →

An online retailer offers two shipping options for heavy packages. The first option charges a flat fee of $15.00$ dollars plus $1.50$ dollars per pound of the package's weight. The second option charges a flat fee of $27.00$ dollars plus $0.70$ dollars per pound of the package's weight. If a package of weight $w$ pounds costs the same total amount of $c$ dollars under both options, what is the value of $w$ ?

1.5

37.5

52.5

15

Show answer & explanation

Answer: 15

Answer

15

To find the weight of the package that yields the same cost under both shipping options, we set the cost expressions equal to each other. The first option is represented by

c = 1.50w + 15

and the second by

c = 0.70w + 27

. Setting them equal gives

1.50w + 15 = 0.70w + 27

. Subtracting

0.70w

from both sides results in

0.80w + 15 = 27

. Subtracting

15

from both sides gives

0.80w = 12

. Dividing by

0.80

yields

w = 15

. Therefore, the weight of the package is 15 pounds.

Step-by-Step Solution

1

Set up the linear equations representing the total cost c for each shipping option based on the package's weight w.

The first option is represented by

c = 1.50w + 15

, and the second option is represented by

c = 0.70w + 27

.

This models the real-world scenario into a system of two linear equations with two variables.

2

Set the two expressions for the total cost c equal to each other to solve for the weight w.

1.50w + 15 = 0.70w + 27

Since the package costs the same under both options, the cost values are equal at this specific weight.

3

Isolate the variable w by subtracting

0.70w

and

15

from both sides of the equation.

0.80w = 12

, which simplifies to

w = 15

.

Subtracting

0.70w

from both sides isolates the variable terms on the left, and subtracting

15

isolates the constants on the right. Dividing by

0.80

solves for w.

Key Concept

Solving a system of linear equations by setting the expressions equal to find the point where two rates intersect.

Estimated Time:1m 30s

Question 363Question →

A chemist cools a liquid compound in a laboratory refrigerator. The temperature of the liquid decreases at a constant rate. After $4$ minutes of cooling, the temperature of the liquid is $22^\circ\text{C}$ . After $10$ minutes of cooling, the temperature of the liquid is $13^\circ\text{C}$ . Which of the following equations represents the relationship between the temperature, $T$ , in degrees Celsius, of the liquid and the number of minutes, $m$ , it has been cooling?

2m + 3T = 74

T = 4m + 22

3m + 2T = 56

2T - 3m = 56

Show answer & explanation

Answer: $3m + 2T = 56$

Answer

The equation

3m + 2T = 56

correctly represents the relationship between the temperature

T

and the cooling time

m

.

The relationship between the temperature

T

and the cooling time

m

is linear, which can be represented by the equation

T = -1.5m + 28

. Rearranging this equation by adding

1.5m

to both sides yields

1.5m + T = 28

. Multiplying the entire equation by

2

to clear the decimal gives

3m + 2T = 56

, which is the correct linear relationship.

Step-by-Step Solution

1

Calculate the rate of temperature change (slope,

k

) per minute of cooling.

k = \frac{13 - 22}{10 - 4} = \frac{-9}{6} = -1.5^\circ\text{C}

per minute.

A linear relationship has a constant rate of change, which is found by dividing the change in temperature by the change in time.

2

Use the point-slope form to write the equation of the line.

T - 22 = -1.5(m - 4)

, which simplifies to

T = -1.5m + 28

.

This models the relationship between

T

and

m

by using one of the given data points and the calculated slope.

3

Rearrange the equation into standard form with integer coefficients.

1.5m + T = 28 \implies 3m + 2T = 56

.

The options are written in standard form, so adding

1.5m

to both sides and multiplying the entire equation by

2

matches the target format.

Key Concept

Linear Equations in Two Variables

Question 364Question →

A linear function $f$ is defined such that its graph passes through the points $(2, 5)$ and $(4, 11)$ . The graph of another linear function, $g$ , is perpendicular to the graph of $f$ . If the graph of $g$ passes through the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of the graph of $g$ ?

Show answer & explanation

Answer: 33

Answer

The correct answer is

33

.

The slope of the linear function

f

is found using the two given points:

\frac{11 - 5}{4 - 2} = 3

. Because the graph of the linear function

g

is perpendicular to the graph of

f

, its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of

g

is

y - 10 = -\frac{1}{3}(x - 3)

, which simplifies to

y = -\frac{1}{3}x + 11

. The

x

-intercept is the value of

x

when

y = 0

. Setting

y

to

0

gives

0 = -\frac{1}{3}x + 11

, which yields

x = 33

.

Step-by-Step Solution

1

Calculate the slope of the linear function

f

The slope of

f

is

3

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Substituting

(2, 5)

and

(4, 11)

yields

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

.

2

Determine the slope of the perpendicular linear function

g

The slope of

g

is

-\frac{1}{3}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The negative reciprocal of

3

is

-\frac{1}{3}

.

3

Write the equation representing function

g

y = -\frac{1}{3}x + 11

Using the point-slope equation

y - y_1 = m(x - x_1)

with the slope

m = -\frac{1}{3}

and the point

(3, 10)

, we get

y - 10 = -\frac{1}{3}(x - 3)

. Distributing and simplifying yields

y - 10 = -\frac{1}{3}x + 1 \implies y = -\frac{1}{3}x + 11

.

4

Find the

x

-intercept of the graph of

g

33

The

x

-intercept is the point on the graph where

y = 0

. Substituting

0

for

y

in the equation of

g

gives

0 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33

.

Key Concept

The relationship between the slopes of perpendicular lines and finding intercepts of linear functions.

Estimated Time:1m 30s

Question 365Question →

A construction company rents a concrete mixer. The total cost, in dollars, of renting the concrete mixer is a linear function of the rental time, in hours. The table below shows some values of the rental time and the corresponding total cost.

Time (hours)	Total cost (dollars)
$3$	$54$
$5$	$78$
$8$	$114$

Which of the following is the total cost, in dollars, of renting the concrete mixer for $12$ hours?

55

144

162

168

Show answer & explanation

Answer: $162$

Answer

162

dollars

The cost function is linear, represented by

C(t) = mt + b

. The rate of change (slope) is calculated as

m = \frac{78 - 54}{5 - 3} = 12

dollars per hour. Using the point

(3, 54)

, we solve for the initial fee (y-intercept)

b

:

54 = 12(3) + b

, which gives

b = 18

. Thus, the linear model is

C(t) = 12t + 18

. For a rental time of

12

hours, the total cost is

C(12) = 12(12) + 18 = 162

dollars.

Step-by-Step Solution

1

Calculate the rate of change (slope) of the linear cost function.

m = 12

Since the relationship is linear, the rate of change is constant. Using the coordinate points

(3, 54)

and

(5, 78)

from the table, the slope is calculated as

m = \frac{78 - 54}{5 - 3} = 12

.

2

Determine the y-intercept (initial fee) of the function.

b = 18

Substitute one of the points, such as

(3, 54)

, and the slope

m = 12

into the slope-intercept form equation

C(t) = mt + b

. This gives

54 = 12(3) + b \Rightarrow 54 = 36 + b \Rightarrow b = 18

.

3

Evaluate the cost function for a rental time of

12

hours.

C(12) = 162

Using the complete linear equation

C(t) = 12t + 18

, substitute

t = 12

to find the total cost:

C(12) = 12(12) + 18 = 144 + 18 = 162

.

Key Concept

Finding and evaluating a linear function from tabular data

Estimated Time:1m 30s

Question 366Question →

A manufacturing plant uses two types of machines, Machine X and Machine Y, to produce a certain component. Machine X produces $15$ components per hour, and Machine Y produces $25$ components per hour. On a certain day, both machines were operated for a combined total of $14$ hours, producing a total of $270$ components. For how many hours was Machine X operated?

Show answer & explanation

Answer: 8

Answer

Machine X was operated for 8 hours.

By setting up the system of linear equations where

x

represents the hours Machine X operated and

y

represents the hours Machine Y operated, we get

x + y = 14

and

15x + 25y = 270

. Substituting

y = 14 - x

into the second equation yields

15x + 25(14 - x) = 270

. Expanding and simplifying results in

-10x + 350 = 270

, which solves to

x = 8

. Thus, Machine X operated for 8 hours.

Step-by-Step Solution

1

Define variables for the hours each machine operated and set up the system of linear equations.

Let

x

be the hours Machine X was operated and

y

be the hours Machine Y was operated. The system is:

x + y = 14

15x + 25y = 270

This translates the verbal descriptions in the problem into algebraic expressions and equations.

2

Express

y

in terms of

x

from the first equation and substitute it into the second equation.

y = 14 - x

15x + 25(14 - x) = 270

Substitution reduces the system from two variables to a single equation in terms of

x

.

3

Solve the linear equation for

x

by distributing the constant and combining like terms.

15x + 350 - 25x = 270

-10x + 350 = 270

-10x = -80

x = 8

Isolating

x

gives the specific number of hours Machine X was operated.

Key Concept

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

Alternative Method

We can solve the system by multiplying the first equation by 25 to get

25x + 25y = 350

. Subtracting the second equation

15x + 25y = 270

from this yields

10x = 80

, which directly gives

x = 8

.

Estimated Time:1m 30s

Question 367Question →

A digital printing press uses paper from a large roll. The total weight $W$ , in pounds, of the paper remaining on the roll after $m$ minutes of continuous printing is given by the equation $W = 120 - 1.5m$ . According to the equation, after how many minutes of continuous printing will the weight of the remaining paper on the roll be exactly $45$ pounds?

Show answer & explanation

Answer: 50

Answer

The correct answer is 50.

To find the number of minutes,

m

, when the remaining weight is

45

pounds, substitute

W = 45

into the given equation

W = 120 - 1.5m

. This yields the equation

45 = 120 - 1.5m

. Subtracting

120

from both sides gives

-75 = -1.5m

. Dividing both sides by

-1.5

results in

m = 50

. Therefore, after

50

minutes, the remaining weight of the paper on the roll is

45

pounds.

Step-by-Step Solution

1

Substitute the target weight into the linear equation.

45 = 120 - 1.5m

We are given that the weight of the remaining paper,

W

, is

45

pounds, and we need to solve for the time in minutes,

m

.

2

Isolate the variable term by subtracting

120

from both sides of the equation.

-75 = -1.5m

To solve for

m

, we first need to isolate the term containing the variable on one side of the equation.

3

Divide both sides by

-1.5

to find the value of

m

.

m = 50

Dividing by the coefficient of the variable completes the isolation and gives the final value.

Key Concept

Interpreting values and solving linear equations in context

Alternative Method

Alternatively, you can determine the total weight of paper consumed:

120 - 45 = 75

pounds. Since the paper is used at a rate of

1.5

pounds per minute, the time required is

\frac{75}{1.5} = 50

minutes.

Estimated Time:1m 30s

Question 368Question →

An artist creates custom ceramic tiles in two shapes: square tiles and hexagonal tiles. Each square tile has an area of $12$ square inches, and each hexagonal tile has an area of $28$ square inches. The artist is designing a mosaic with a total area of $460$ square inches using only these two types of tiles. If the artist uses $15$ square tiles, how many hexagonal tiles are used in the mosaic?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10. By setting up a linear equation representing the total area of the mosaic,

12s + 28h = 460

, and substituting the given value of 15 for the number of square tiles, we solve for the number of hexagonal tiles to get 10.

To find the number of hexagonal tiles used in the mosaic, we set up a linear equation in two variables representing the total area. Let

s

represent the number of square tiles, and let

h

represent the number of hexagonal tiles. The total area is the sum of the areas of the square tiles and the hexagonal tiles, which is

12s + 28h = 460

. Given that the artist uses

15

square tiles, we substitute

s = 15

into the equation to get

12(15) + 28h = 460

. Simplifying the equation yields

180 + 28h = 460

. Subtracting

180

from both sides gives

28h = 280

. Dividing both sides by

28

yields

h = 10

. Therefore, the artist uses 10 hexagonal tiles.

Step-by-Step Solution

1

Define variables and set up the linear equation representing the total area of the mosaic.

12s + 28h = 460

, where

s

is the number of square tiles and

h

is the number of hexagonal tiles.

This models the relationship between the quantities of each tile type used and the total surface area of the mosaic.

2

Substitute the given number of square tiles into the equation.

12(15) + 28h = 460

, which simplifies to

180 + 28h = 460

.

We are given that the artist uses exactly 15 square tiles in the design.

3

Isolate the variable term by subtracting the constant from both sides.

28h = 280

.

This isolates the term containing the unknown variable on one side of the equation.

4

Solve for the remaining variable by dividing both sides by its coefficient.

h = 10

.

Dividing both sides by the coefficient yields the final count of hexagonal tiles.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 369Question →

A company uses a fleet of refrigerated delivery trucks to transport fresh produce. The temperature inside a truck's cooling compartment, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), $t$ minutes after the cooling system is turned on is modeled by the equation below:

T = 68 - 1.25t

Where $0 \leq t \leq 24$ . Which of the following is the best interpretation of the number $68$ in this context?

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

The rate, in degrees Fahrenheit per minute, at which the temperature inside the compartment decreases

The temperature inside the compartment, in degrees Fahrenheit, 1 minute after the cooling system is turned on, which is

67.25^\circ\text{F}

The number of minutes it takes for the temperature inside the compartment to reach

0^\circ\text{F}

Show answer & explanation

Answer: The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

Answer

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

The correct answer correctly identifies the meaning of the constant term

68

as the y-intercept of the linear equation. In this context, the y-intercept represents the value of

T

when the time

t

is

0

, which corresponds to the initial temperature inside the compartment before any cooling occurs.

Step-by-Step Solution

1

Identify the form of the linear equation and its components.

The equation is in the slope-intercept form,

T = mt + b

, where

m = -1.25

is the slope and

b = 68

is the

T

-intercept (or vertical intercept).

Recognizing the mathematical meaning of each coefficient is the first step in interpreting them in a real-world context.

2

Determine the physical meaning of the vertical intercept (

b = 68

).

The vertical intercept occurs when the independent variable,

t

(time in minutes), is equal to

0

. Evaluating

T

at

t = 0

gives

T(0) = 68 - 1.25(0) = 68

.

Setting the independent variable to zero allows us to define the starting state of the system.

3

Interpret this starting state in the context of the problem.

At

t = 0

, the cooling system has just been turned on. Thus,

68^\circ\text{F}

represents the temperature inside the truck's cooling compartment when the system is first turned on.

Connecting the value of

T(0)

back to the scenario provides the correct interpretation of the constant

68

.

Key Concept

Interpreting the y-intercept of a linear model in context

Estimated Time:1m 30s

Question 370Question →

Given the system of equations below:

\begin{aligned} 3x - y &= 11 \\ 2x + 3y &= 11 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + 2y$ ?

4

6

8

9

Show answer & explanation

Answer: $6$

Answer

The value of

x + 2y

is

6

.

To solve the system, we can isolate

y

in the first equation to get

y = 3x - 11

. Substituting this into the second equation gives

2x + 3(3x - 11) = 11

, which simplifies to

11x - 33 = 11

, or

11x = 44

, resulting in

x = 4

. Substituting

x = 4

back into the equation for

y

gives

y = 3(4) - 11 = 1

. The value of the expression

x + 2y

is therefore

4 + 2(1) = 6

.

Step-by-Step Solution

1

Isolate

y

in the first equation.

y = 3x - 11

To prepare for substitution into the second equation.

2

Substitute the expression for

y

into the second equation and solve for

x

.

2x + 3(3x - 11) = 11 \implies 11x - 33 = 11 \implies 11x = 44 \implies x = 4

To find the value of

x

.

3

Substitute the value of

x

back into the isolated equation for

y

to solve for

y

.

y = 3(4) - 11 \implies y = 1

To find the value of

y

.

4

Evaluate the expression

x + 2y

using the values of

x

and

y

.

4 + 2(1) = 6

To find the final requested value.

Key Concept

Solving a system of linear equations using substitution or elimination, and evaluating a linear combination of the variables.

Alternative Method

Alternatively, the system can be solved using the elimination method. Multiply the first equation by

3

to align the

y

-coefficients:

9x - 3y = 33

. Add this equation to the second equation,

2x + 3y = 11

, to eliminate

y

, yielding

11x = 44

, so

x = 4

. Then substitute

x = 4

into either original equation to find

y = 1

.

Estimated Time:1m 30s

Question 371Question →

A municipal water utility charges customers a flat monthly fee plus a constant rate per thousand gallons of water used. In June, a household used $8$ thousand gallons of water and was charged a total of $\$46.00$ . In July, the same household used $12$ thousand gallons of water and was charged a total of $\$62.00$ . If $x$ represents the number of thousand gallons of water used in a month, and $y$ represents the total monthly charge, in dollars, which of the following equations represents the relationship between $x$ and $y$ ?

y = 0.25x + 44

y = 14x + 4

y = 4x + 14

y = 4x + 78

Show answer & explanation

Answer: $y = 4x + 14$

Answer

y = 4x + 14

The correct equation is

y = 4x + 14

. The constant rate of change (slope) is found by dividing the difference in monthly charges by the difference in water usage:

\frac{62 - 46}{12 - 8} = 4

. Using the point-slope form or slope-intercept form with the point

(8, 46)

yields

46 = 4(8) + b

, which simplifies to

b = 14

for the y-intercept (the flat monthly fee). Substituting these parameters back into the slope-intercept form yields the correct relationship.

Step-by-Step Solution

1

Calculate the constant rate of change (slope,

m

) using the two given data points,

(8, 46)

and

(12, 62)

.

m = \frac{62 - 46}{12 - 8} = \frac{16}{4} = 4

.

The constant rate of change represents the slope of the linear equation.

2

Use the slope-intercept form,

y = mx + b

, and substitute one of the data points, such as

(8, 46)

, to solve for the y-intercept (

b

).

46 = 4(8) + b \implies 46 = 32 + b \implies b = 14

.

Substituting a known point allows us to isolate and find the value of the flat monthly fee, which is the y-intercept.

3

Write the final equation by substituting the calculated slope (

m = 4

) and y-intercept (

b = 14

) back into the slope-intercept form.

y = 4x + 14

.

This combines the rate per thousand gallons and the flat fee into the final linear relationship.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 372Question →

Due to rising global temperatures, the thickness $T$ , in meters, of a mountain glacier is decreasing at a constant rate. A researcher models the thickness of the glacier using the equation $T = 48.5 - 0.12w$ , where $w$ represents the number of weeks since the start of a monitoring study. According to the model, by how many meters does the thickness of the glacier decrease every $50$ weeks?

Show answer & explanation

Answer: 6

Answer

The glacier's thickness decreases by 6 meters.

In the linear model

T = 48.5 - 0.12w

, the thickness

T

decreases by

0.12

meters for each increase of

1

in

w

(each week). Therefore, over a period of

50

weeks, the total decrease in thickness is

0.12 \times 50 = 6

meters.

Step-by-Step Solution

1

Identify the weekly rate of decrease from the slope of the equation.

0.12 meters per week

In the linear model

T = 48.5 - 0.12w

, the coefficient of

w

is

-0.12

. This slope represents the rate of change, indicating a decrease of

0.12

meters in thickness per week.

2

Calculate the cumulative decrease over 50 weeks.

6

Multiply the weekly rate of decrease (

0.12

meters per week) by the duration (

50

weeks) to obtain the total decrease:

0.12 \times 50 = 6

.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 30s

Question 373Question →

\begin{aligned} 4x - 3y &= 25 \\ -2x + 5y &= -9 \end{aligned}

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

Show answer & explanation

Answer: 6

Answer

6

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to align the coefficients of

x

:

2(-2x + 5y) = 2(-9) \implies -4x + 10y = -18

Next, add this new equation to the first equation to eliminate

x

:

(4x - 3y) + (-4x + 10y) = 25 + (-18)

7y = 7

y = 1

Substitute

y = 1

back into the second equation to solve for

x

:

-2x + 5(1) = -9

-2x + 5 = -9

-2x = -14

x = 7

Finally, calculate the value of the requested expression

x - y

:

x - y = 7 - 1 = 6

Thus, the correct response is 6.

Step-by-Step Solution

1

Multiply the second equation by 2 to prepare for the elimination of

x

.

-4x + 10y = -18

This creates coefficients for

x

in both equations that are additive opposites, allowing

x

to be eliminated when the equations are added.

2

Add the first equation and the modified second equation together to solve for

y

.

7y = 7

, which simplifies to

y = 1

Adding the equations eliminates the

x

terms and leaves a single-variable equation in terms of

y

.

3

Substitute

y = 1

back into one of the original equations to solve for

x

.

-2x + 5(1) = -9

, which simplifies to

-2x = -14

, yielding

x = 7

Now that the value of

y

is known, it can be substituted into either equation to find the corresponding value of

x

.

4

Calculate the value of the expression

x - y

.

7 - 1 = 6

The question asks specifically for the value of the difference

x - y

, so we subtract the value of

y

from the value of

x

.

Key Concept

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Estimated Time:1m 30s

Question 374Question →

In the $xy$ -plane, the graph of the linear equation $kx - 3y = 18$ , where $k$ is a constant, has a $y$ -intercept of $(0, b)$ and an $x$ -intercept of $(a, 0)$ , where $a$ and $b$ are constants. If $a - b = 15$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

2

To find the value of

k

, we first determine the intercepts of the linear equation

kx - 3y = 18

. The

y

-intercept

(0, b)

occurs where

x = 0

. Substituting

x = 0

gives

-3b = 18

, which solves to

b = -6

. Next, we use the given relation

a - b = 15

. Substituting

b = -6

yields

a - (-6) = 15

, or

a + 6 = 15

, which gives

a = 9

. The

x

-intercept is therefore

(9, 0)

. Since this point lies on the line, we substitute

x = 9

and

y = 0

into the original equation:

k(9) - 3(0) = 18

. This simplifies to

9k = 18

, which gives

k = 2

.

Step-by-Step Solution

1

Find the

y

-coordinate of the

y

-intercept (

b

) by setting

x = 0

in the equation

kx - 3y = 18

.

b = -6

The

y

-intercept occurs where the graph crosses the

y

-axis, which corresponds to

x = 0

.

2

Use the equation

a - b = 15

and the value of

b = -6

to solve for

a

.

a = 9

Substituting

b = -6

into

a - b = 15

gives

a - (-6) = 15

, which simplifies to

a + 6 = 15

.

3

Find the value of

k

by substituting the

x

-intercept

(9, 0)

into the equation

kx - 3y = 18

.

k = 2

Since

(a, 0) = (9, 0)

is the

x

-intercept, it must satisfy the equation of the line.

Key Concept

Determining intercepts of a linear equation in two variables and using them to find unknown constants.

Question 375Question →

An agricultural drone is used to spray liquid fertilizer on a field. The volume of fertilizer remaining in the drone's tank, $F$ , in liters, can be modeled by the equation $F = 120 - 4.8t$ , where $t$ is the number of minutes after the drone begins spraying. What is the best interpretation of the number $4.8$ in this context?

The volume of fertilizer, in liters, in the drone's tank before the spraying operation begins.

The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

The number of minutes it takes for the drone to spray one liter of fertilizer.

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

Show answer & explanation

Answer: The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

Answer

The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

In the linear equation

F = 120 - 4.8t

, the variable

F

represents the remaining fertilizer in liters, and

t

represents the elapsed time in minutes. The value 4.8 is the coefficient of

t

, which represents the rate of change of the fertilizer in the tank. The negative sign in front of 4.8 indicates a decrease over time, meaning that the drone sprays 4.8 liters of fertilizer from the tank each minute.

Step-by-Step Solution

1

Identify the role of the constant 4.8 in the given linear equation

F = 120 - 4.8t

.

The coefficient of the independent variable

t

is

-4.8

, which represents the slope of the linear relationship.

In the slope-intercept form of a linear equation, the coefficient of the variable represents the rate of change (slope).

2

Interpret the meaning of the slope in the context of the problem.

Since

F

is measured in liters and

t

is measured in minutes, the slope represents a change in liters per minute. The negative sign indicates that the volume of fertilizer is decreasing.

The rate of change is the change in the dependent variable (liters of fertilizer) divided by the change in the independent variable (minutes).

3

Select the option that correctly describes this rate of change.

The rate of decrease is 4.8 liters per minute, which means 4.8 liters of fertilizer are sprayed from the tank each minute.

This matches the definition of the rate at which fertilizer is leaving the tank.

Key Concept

Interpreting the slope of a linear equation in a real-world context.

Question 376Question →

\begin{aligned} k x - 5y &= 8 \\ 3x - 2y &= 12 \end{aligned}

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

\frac{6}{5}

\frac{10}{3}

\frac{15}{2}

-\frac{15}{2}

Show answer & explanation

Answer: $\frac{15}{2}$

Answer

\frac{15}{2}

The correct answer is the value that makes the two lines parallel. Rewriting both equations in slope-intercept form gives the slopes

\frac{k}{5}

and

\frac{3}{2}

. Equating these slopes yields

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

, which simplifies to

\frac{15}{2}

. Since their

y

-intercepts (

-\frac{8}{5}

and

-6

) are distinct, the lines are parallel and never intersect, meaning the system has no solution.

Step-by-Step Solution

1

Convert the first equation to slope-intercept form (

y = mx + b

).

kx - 5y = 8 \implies -5y = -kx + 8 \implies y = \frac{k}{5}x - \frac{8}{5}

This identifies the slope (

m_1 = \frac{k}{5}

) and the

y

-intercept (

b_1 = -\frac{8}{5}

) of the first line.

2

Convert the second equation to slope-intercept form.

3x - 2y = 12 \implies -2y = -3x + 12 \implies y = \frac{3}{2}x - 6

This identifies the slope (

m_2 = \frac{3}{2}

) and the

y

-intercept (

b_2 = -6

) of the second line.

3

Equate the two slopes to find the value of

k

that makes the lines parallel.

\frac{k}{5} = \frac{3}{2} \implies k = 5 \times \frac{3}{2} = \frac{15}{2}

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

y

-intercepts). Since the

y

-intercepts

-\frac{8}{5}

and

-6

are different, setting the slopes equal guarantees no solution.

Key Concept

A system of two linear equations has no solution if the lines represented by the equations have the same slope but different

y

-intercepts (parallel lines).

Estimated Time:1m 30s

Question 377Question →

The table below shows the remaining balance, $B$ , in dollars, on a transit card after a user has taken $r$ rides on a city bus.

Number of rides ( $r$ )	Remaining balance ( $B$ , in dollars)
$3$	$22.50$
$8$	$10.00$
$12$	$0.00$

Which of the following equations represents the relationship between $B$ and $r$ ?

B = 22.50 - 2.50r

B = 30.00 - 0.40r

B = 30.00 - 2.50r

B = 30.00 + 2.50r

Show answer & explanation

Answer: $B = 30.00 - 2.50r$

Answer

B = 30.00 - 2.50r

The correct equation has a rate of change of

-2.50

dollars per ride and a starting balance of

30.00

dollars. The rate of change (slope) can be calculated by finding the change in balance divided by the change in the number of rides between two points, such as

(3, 22.50)

and

(8, 10.00)

, which gives

\frac{10.00 - 22.50}{8 - 3} = -2.50

. Using the slope-intercept form

B = mr + B_0

and the point

(12, 0.00)

, we get

0.00 = -2.50(12) + B_0

, which simplifies to

B_0 = 30.00

. Therefore, the equation is

B = 30.00 - 2.50r

.

Step-by-Step Solution

1

Calculate the slope (rate of change) using two points from the table, such as

(3, 22.50)

and

(8, 10.00)

.

The slope is

m = \frac{10.00 - 22.50}{8 - 3} = \frac{-12.50}{5} = -2.50

.

The slope represents the constant change in the remaining balance per ride.

2

Use the slope-intercept form

B = mr + B_0

and one point to find the initial balance

B_0

.

Substituting

m = -2.50

and the point

(12, 0.00)

gives

0.00 = -2.50(12) + B_0

, which simplifies to

0.00 = -30.00 + B_0

, so

B_0 = 30.00

.

The initial balance

B_0

is the y-intercept, which is the value of

B

when

r = 0

.

3

Write the final linear equation representing the relationship.

The final equation is

B = 30.00 - 2.50r

.

This equation models the remaining balance on the card as a function of the number of rides taken.

Key Concept

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

Question 378Question →

A municipal water reservoir contains 250 million gallons of water. During a dry spell, water is released from the reservoir at a constant rate of 3.5 million gallons per day. Additionally, water evaporates from the reservoir at a constant rate of 0.3 million gallons per day. If no water enters the reservoir, the total amount of water $W$ , in million gallons, remaining in the reservoir after $d$ days of the dry spell is modeled by the equation $W = 250 - r d$ , where $r$ is a constant. What is the value of $r$ ?

Show answer & explanation

Answer: 3.8

Answer

The correct value of

r

is 3.8.

The constant

r

in the linear equation

W = 250 - r d

represents the total rate, in million gallons per day, at which the water volume in the reservoir decreases. Since water is lost through both release (

3.5

million gallons per day) and evaporation (

0.3

million gallons per day), the total rate of decrease is the sum of these two rates, which is

3.5 + 0.3 = 3.8

million gallons per day. Therefore, the value of

r

is

3.8

.

Step-by-Step Solution

1

Identify the factors causing a decrease in the reservoir's water volume.

Water is lost through release at

3.5

million gallons per day and evaporation at

0.3

million gallons per day.

Both release and evaporation contribute to the total rate of water depletion.

2

Calculate the total daily rate of water loss.

3.5 + 0.3 = 3.8

million gallons per day.

Adding the individual rates of loss yields the overall rate of decrease.

3

Compare the total daily rate of water loss to the model equation

W = 250 - r d

.

r = 3.8

.

In the linear model,

250

represents the initial amount of water, and

r

represents the constant rate at which water decreases per day. Thus,

r

is the total daily rate of water loss.

Key Concept

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

Question 379Question →

A chemist mixes a $10\%$ acid solution and a $30\%$ acid solution to create $200$ milliliters of a mixture that is $18\%$ acid. How many milliliters of the $30\%$ acid solution are in the mixture?

Show answer & explanation

Answer: 80

Answer

80

The correct answer is

80

milliliters. By translating the problem into a system of linear equations, we let

x

be the volume of the

10\%

acid solution and

y

be the volume of the

30\%

acid solution. Since the total volume is

200

milliliters,

x + y = 200

. The amount of acid in the solutions must sum to the amount of acid in the final mixture, so

0.10x + 0.30y = 0.18(200)

, which simplifies to

0.10x + 0.30y = 36

. Multiplying this equation by

10

gives

x + 3y = 360

. Subtracting the first equation from this yields

(x + 3y) - (x + y) = 360 - 200

, or

2y = 160

. Solving for

y

gives

80

milliliters of the

30\%

solution.

Step-by-Step Solution

1

Define variables and set up the system of equations.

Let

x

be the number of milliliters of the

10\%

acid solution and

y

be the number of milliliters of the

30\%

acid solution. The total volume equation is

x + y = 200

. The total acid content equation is

0.10x + 0.30y = 0.18(200)

.

This sets up the system of linear equations representing the physical constraints of the mixture.

2

Simplify the acid content equation and prepare for elimination.

0.10x + 0.30y = 36

. Multiplying the entire equation by

10

yields

x + 3y = 360

.

Eliminating decimals simplifies the coefficients and makes it easier to solve the system using integer arithmetic.

3

Eliminate

x

by subtracting the total volume equation from the simplified acid content equation.

(x + 3y) - (x + y) = 360 - 200 \implies 2y = 160 \implies y = 80

.

This isolates the variable

y

, which directly represents the volume of the

30\%

acid solution requested in the problem.

Key Concept

Solving systems of linear equations in context

Question 380Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the points $(2, 15)$ and $(6, 7)$ . What is the value of $b$ ?

Show answer & explanation

Answer: 19

Answer

The value of

b

is

19

.

The slope of the line is found using the two given points:

m = \frac{7 - 15}{6 - 2} = -2

. Substituting the slope

m = -2

and the point

(2, 15)

into the equation

y = mx + b

gives

15 = -2(2) + b

, which simplifies to

15 = -4 + b

. Adding

4

to both sides yields

b = 19

.

Step-by-Step Solution

1

Calculate the slope of the line passing through

(2, 15)

and

(6, 7)

.

m = \frac{7 - 15}{6 - 2} = \frac{-8}{4} = -2

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

.

2

Substitute the slope

m = -2

and the coordinates of one point, such as

(2, 15)

, into the equation

y = mx + b

to find

b

.

15 = -2(2) + b \implies 15 = -4 + b

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

3

Solve the equation for

b

.

b = 19

Add

4

to both sides of the equation to isolate the variable

b

.

Key Concept

Determining the equation of a line given two points.

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