Algebra Practice Questions — SAT — Page 18

Question 341Question →

A ride-share driver uses a mobile application to track their daily net earnings. The driver's net earnings, $E$ , in dollars, after completing $n$ rides in a day can be modeled by the equation $E = 18n - 25$ . Which of the following is the best interpretation of the number $25$ in this context?

The driver's net earnings, in dollars, for each completed ride.

The driver's daily starting cost, in dollars, before any rides are completed.

The driver's net earnings, in dollars, after completing exactly one ride.

The number of completed rides for which the driver's net earnings are equal to 0 dollars.

Show answer & explanation

Answer: The driver's daily starting cost, in dollars, before any rides are completed.

Answer

The driver's daily starting cost, in dollars, before any rides are completed.

In the linear equation

E = 18n - 25

, the variable

E

represents net earnings and

n

represents the number of rides. The constant term

-25

represents the value of

E

when

n = 0

. In this context, starting with

-25

dollars in net earnings means the driver has a daily starting cost or fee of

25

dollars before completing any rides.

Step-by-Step Solution

1

Identify the component of the linear equation

E = 18n - 25

associated with the number

25

.

The number

25

is part of the constant term

-25

in the equation.

Linear equations in context are typically written in the form

y = mx + b

, where

m

is the slope and

b

is the y-intercept.

2

Determine the contextual meaning of the constant term (y-intercept) in this model.

The constant term

-25

represents the value of

E

when

n = 0

.

Setting the independent variable

n

(number of rides) to

0

gives the starting value of the dependent variable

E

(net earnings).

3

Interpret the negative value in terms of daily operations.

An initial net earning of

-25

dollars means the driver starts with a cost or loss of

25

dollars.

A negative starting value indicates a cost or fee that must be paid before any revenue is generated.

Key Concept

Interpreting the y-intercept of a linear model in context.

Question 342Question →

The table below shows several values of $x$ and the corresponding values for the linear functions $f$ and $g$ .

$x$	$f(x)$	$g(x)$
1	7	17
2	9	14
4	13	8

If $(x, y)$ is the solution to the system of equations formed by $y = f(x)$ and $y = g(x)$ , what is the value of $2x + y$ ?

5

17

15

25

Show answer & explanation

Answer: 17

Answer

The correct value of the expression is 17.

The correct answer is the value obtained by finding the linear equations for both functions, solving the resulting system of equations, and substituting the coordinates into the expression. The equation for the first function is

y = 2x + 5

and the equation for the second function is

y = -3x + 20

. Equating the two expressions yields

2x + 5 = -3x + 20

, which simplifies to

5x = 15

or

x = 3

. Substituting

x = 3

back into either equation yields

y = 11

. Evaluating the expression

2x + y

with these values gives

2(3) + 11 = 17

.

Step-by-Step Solution

1

Determine the linear equation for

f(x)

using the points in the table.

The slope of

f(x)

is

\frac{9 - 7}{2 - 1} = 2

. Using the point-slope form with

(1, 7)

, the equation is

y - 7 = 2(x - 1)

, which simplifies to

y = 2x + 5

.

Finding the equation of the first line is necessary to set up the system of linear equations.

2

Determine the linear equation for

g(x)

using the points in the table.

The slope of

g(x)

is

\frac{14 - 17}{2 - 1} = -3

. Using the point-slope form with

(1, 17)

, the equation is

y - 17 = -3(x - 1)

, which simplifies to

y = -3x + 20

.

Finding the equation of the second line completes the system of equations.

3

Find the intersection point of

y = f(x)

and

y = g(x)

by setting the two equations equal to each other.

2x + 5 = -3x + 20 \Rightarrow 5x = 15 \Rightarrow x = 3

. Substituting

x = 3

into

y = 2x + 5

gives

y = 2(3) + 5 = 11

.

Solving the system of equations yields the values of

x

and

y

at the intersection point.

4

Calculate the value of

2x + y

using the solution

(3, 11)

.

2(3) + 11 = 6 + 11 = 17

.

This evaluates the specific linear combination requested in the question.

Key Concept

Solving systems of linear equations by translating tabular data into linear equations and finding their point of intersection.

Question 343Question →

An agricultural drone is spraying liquid fertilizer on a field at a constant rate. The total amount of fertilizer remaining in the drone's tank, in liters, can be modeled by a linear function of the time, in minutes, since the drone began spraying. After $3$ minutes of spraying, $84$ liters of fertilizer remain in the tank. After $8$ minutes of spraying, $54$ liters of fertilizer remain in the tank. How many minutes of spraying will it take for the tank to become completely empty?

Show answer & explanation

Answer: 17

Answer

17

The correct answer is 17. The volume of fertilizer in the tank decreases at a constant rate of 6 liters per minute. Since 84 liters remain after 3 minutes, the initial volume of fertilizer in the tank was 102 liters. Dividing the initial volume of 102 liters by the rate of 6 liters per minute yields 17 minutes for the tank to be completely empty.

Step-by-Step Solution

1

Calculate the constant rate of change (slope) of the remaining fertilizer volume.

-6 liters per minute

The slope formula is used with the two data points representing time and volume:

(3, 84)

and

(8, 54)

.

2

Determine the initial volume of fertilizer in the tank (y-intercept).

102 liters

Using the slope-intercept form

F(t) = mt + b

, we substitute the slope

m = -6

and the point

(3, 84)

to solve for

b

.

3

Find the time when the volume of remaining fertilizer reaches 0.

17 minutes

Setting the linear function equal to 0 and solving for time gives the total duration until the tank is empty.

Key Concept

Linear Functions and Graphs

Question 344Question →

At a local farmer's market, a vendor sells two types of fruit baskets: a standard basket and a deluxe basket. The standard basket contains $3$ apples and $2$ oranges. The deluxe basket contains $5$ apples and $4$ oranges. If a customer bought a combination of these baskets containing a total of $31$ apples and $22$ oranges, what is the total number of baskets the customer bought?

Show answer & explanation

Answer: 9

Answer

The total number of baskets the customer bought is 9.

The correct answer is obtained by setting up a system of equations where

s

is the number of standard baskets and

d

is the number of deluxe baskets. The equations

3s + 5d = 31

and

2s + 4d = 22

represent the totals of apples and oranges respectively. Dividing the second equation by 2 gives

s + 2d = 11

, which can be rearranged to

s = 11 - 2d

. Substituting this expression into the first equation yields

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, meaning

d = 2

. Substituting

d = 2

back gives

s = 7

. The total number of baskets is the sum of both types,

7 + 2 = 9

.

Step-by-Step Solution

1

Define variables and set up the system of linear equations.

Let

s

be the number of standard baskets and

d

be the number of deluxe baskets. The equations are:

3s + 5d = 31

(for apples) and

2s + 4d = 22

(for oranges).

To represent the relationships between the number of baskets and the total quantities of fruits mathematically.

2

Solve the system of equations using substitution.

From the second equation, dividing by 2 yields

s + 2d = 11

, so

s = 11 - 2d

. Substituting this into the first equation gives

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, leading to

d = 2

.

To find the number of deluxe baskets purchased.

3

Calculate the number of standard baskets and sum the two counts to find the total number of baskets.

Substitute

d = 2

into

s = 11 - 2d

to get

s = 7

. The total number of baskets is

s + d = 7 + 2 = 9

.

To answer the specific question asking for the total number of baskets bought.

Key Concept

Systems of Linear Equations

Question 345Question →

In the $xy$ -plane, the line with equation $ax + 4y = 36$ , where $a$ is a constant, has a $y$ -intercept of $(0, p)$ and an $x$ -intercept of $(q, 0)$ , where $p$ and $q$ are positive integers. If $p + q = 15$ , what is the value of $a$ ?

Show answer & explanation

Answer: 6

Answer

6

To find the value of

a

, we first determine the

y

-intercept of the line by setting

x = 0

in the equation

ax + 4y = 36

. This gives

4y = 36

, so

y = 9

. Thus, the

y

-intercept is

(0, 9)

, which means

p = 9

. Using the given relationship

p + q = 15

, we substitute

p = 9

to find

q = 6

. The

x

-intercept is therefore

(6, 0)

. Substituting these coordinates back into the line's equation gives

a(6) + 4(0) = 36

, which simplifies to

6a = 36

. Solving for

a

yields

a = 6

.

Step-by-Step Solution

1

Set

x = 0

in the equation

ax + 4y = 36

to find the

y

-intercept.

4y = 36 \implies y = 9

, so

p = 9

.

The

y

-intercept of a graph is the point where

x = 0

.

2

Substitute

p = 9

into the equation

p + q = 15

to solve for

q

.

9 + q = 15 \implies q = 6

.

We are given that the sum of the

y

-coordinate of the

y

-intercept and the

x

-coordinate of the

x

-intercept is

15

.

3

Substitute the

x

-intercept

(6, 0)

into the equation

ax + 4y = 36

to solve for

a

.

a(6) + 4(0) = 36 \implies 6a = 36 \implies a = 6

.

Since the

x

-intercept is

(q, 0)

and

q = 6

, the point

(6, 0)

must satisfy the equation of the line.

Key Concept

Finding and using intercepts of a linear equation in two variables.

Question 346Question →

A botanist models the rate of water transpiration of a plant species under various temperature conditions. The table below shows the estimated transpiration rate, $R$ , in milligrams of water per square decimeter of leaf area per hour ( $\text{mg}/(\text{dm}^2\cdot\text{h})$ ), at various ambient temperatures, $t$ , in degrees Celsius ( $^\circ\text{C}$ ).

Temperature ( $t$ )	Transpiration rate ( $R$ )
$15$	$84.5$
$20$	$108.5$
$25$	$132.5$
$30$	$156.5$

The relationship between the ambient temperature and the transpiration rate is linear. Based on the model, what is the estimated increase in the transpiration rate, in milligrams of water per square meter of leaf area per minute, for each increase of $1$ degree Celsius in the ambient temperature? (Note: $1\text{ square meter} = 100\text{ square decimeters}$ )

Show answer & explanation

Answer: 8

Answer

8

To find the estimated increase in transpiration rate per degree Celsius in the new units, we first determine the rate of change in the original units from the table. The change in temperature is

20 - 15 = 5^\circ\text{C}

, and the corresponding change in transpiration rate is

108.5 - 84.5 = 24\text{ mg}/(\text{dm}^2\cdot\text{h})

. The rate of change is

\frac{24}{5} = 4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

temperature increase. Converting this rate to square meters, we multiply by

100

because

1\text{ square meter} = 100\text{ square decimeters}

, giving

480\text{ mg}/(\text{m}^2\cdot\text{h})

. Finally, to convert to minutes, we divide by

60

because

1\text{ hour} = 60\text{ minutes}

, which yields

\frac{480}{60} = 8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

temperature increase.

Step-by-Step Solution

1

Find the rate of change of the transpiration rate with respect to temperature from the given data table.

Slope =

4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

The rate of change represents the increase in transpiration rate for each

1^\circ\text{C}

increase in temperature in the original units.

2

Convert the unit of area in the rate of change from square decimeters to square meters.

Rate =

480\text{ mg}/(\text{m}^2\cdot\text{h})

per

1^\circ\text{C}

Since

1\text{ square meter} = 100\text{ square decimeters}

, the rate per square meter is

100

times the rate per square decimeter.

3

Convert the unit of time in the rate of change from hours to minutes.

Rate =

8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

Since there are

60

minutes in an hour, dividing the hourly rate by

60

gives the rate per minute.

Key Concept

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

Estimated Time:2m 30s

Question 347Question →

An online retail store charges a flat shipping fee plus a separate fee per package for bulk deliveries. The total shipping cost, $C$ , in dollars, for an order containing $x$ standard packages and $y$ deluxe packages can be modeled by the equation $C = 15x + 10y + 25$ . If the total shipping cost for a certain order is $195$ dollars and the order contains 8 deluxe packages, how many standard packages are in the order?

6

5

4

8

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. Substituting the total cost of

195

for

C

and the 8 deluxe packages for

y

into the equation

C = 15x + 10y + 25

yields

195 = 15x + 10(8) + 25

. Simplifying the equation gives

195 = 15x + 105

. Subtracting 105 from both sides results in

90 = 15x

. Dividing both sides by 15 gives

x = 6

.

Step-by-Step Solution

1

Substitute the given values into the linear equation.

Substituting

C = 195

and

y = 8

into the equation

C = 15x + 10y + 25

yields

195 = 15x + 10(8) + 25

.

This sets up the equation with only one variable,

x

, which we need to solve for.

2

Simplify the constants on the right side of the equation.

195 = 15x + 80 + 25

, which simplifies to

195 = 15x + 105

.

Combining like terms simplifies the algebraic expression.

3

Isolate the variable term

15x

.

Subtracting 105 from both sides of the equation gives

15x = 195 - 105

, which simplifies to

15x = 90

.

To solve for

x

, we must isolate the term containing the variable on one side.

4

Solve for the variable

x

.

x = \frac{90}{15} = 6

.

Dividing by the coefficient of

x

isolates the variable completely.

Key Concept

Solving linear equations in two variables by substituting known values and isolating the variable.

Question 348Question →

A commercial hydroponic farm monitors the volume of nutrient solution in a reservoir. The table below shows the volume of solution remaining, $V$ , in liters, after $t$ hours of operation.

Time ( $t$ , hours)	Volume ( $V$ , liters)
0	450
10	432
20	414
30	396

The relationship between the time and the volume of remaining solution can be modeled by a linear equation. Which of the following is the best interpretation of the slope of the graph of this equation in the $tV$ -plane?

The nutrient solution is being depleted at a rate of 1.8 liters per hour.

The initial volume of the nutrient solution in the reservoir is 450 liters.

The reservoir will be completely empty after 250 hours.

The volume of the nutrient solution decreases by 1 liter every 1.8 hours.

Show answer & explanation

Answer: The nutrient solution is being depleted at a rate of 1.8 liters per hour.

Answer

The nutrient solution is being depleted at a rate of 1.8 liters per hour.

The slope of a linear function represents the constant rate of change of the dependent variable with respect to the independent variable. By selecting two data points from the table, such as

(0, 450)

and

(10, 432)

, the slope is calculated as

\frac{432 - 450}{10 - 0} = -1.8

. Since the volume

V

is measured in liters and the time

t

is in hours, this slope represents a decrease of

1.8

liters of nutrient solution per hour of operation. Therefore, the nutrient solution is being depleted at a rate of

1.8

liters per hour.

Step-by-Step Solution

1

Calculate the slope using two points from the table.

Using the points

(0, 450)

and

(10, 432)

, the slope is

\frac{432 - 450}{10 - 0} = \frac{-18}{10} = -1.8

.

The slope of a linear relationship is given by the change in the dependent variable (Volume) divided by the change in the independent variable (Time).

2

Interpret the unit of the slope in context.

The unit of the slope is liters per hour (

\text{L/h}

), representing a change of

-1.8

liters for every hour of operation.

Determining the units of the rate helps verify what physical quantity the slope represents.

3

Relate the negative sign of the slope to the context of depletion.

A negative slope of

-1.8

means the volume of nutrient solution decreases by

1.8

liters each hour, which translates to a depletion rate of

1.8

liters per hour.

Connecting the mathematical sign to the real-world action clarifies whether the quantity is increasing or decreasing.

Key Concept

Interpreting the slope of a linear equation as a constant rate of change in a real-world context.

Estimated Time:1m 30s

Question 349Question →

In a certain video game, players earn points by completing daily quests and weekly challenges. Last month, a player completed a total of $24$ activities, which consisted of daily quests and weekly challenges, and earned a total of $190$ points. Each daily quest completed was worth $5$ points, and each weekly challenge completed was worth $15$ points. How many weekly challenges did the player complete?

5

7

9

17

Show answer & explanation

Answer: $7$

Answer

The player completed

7

weekly challenges.

To find the number of weekly challenges, we set up a system of linear equations. Let

d

be the number of daily quests and

w

be the number of weekly challenges. We have

d + w = 24

and

5d + 15w = 190

. Expressing

d

as

24 - w

and substituting it into the second equation gives

5(24 - w) + 15w = 190

, which simplifies to

120 - 5w + 15w = 190

. Combining like terms yields

120 + 10w = 190

, so

10w = 70

, which gives

w = 7

. This matches the correct value.

Step-by-Step Solution

1

Represent the situation with a system of linear equations.

Let

d

represent the number of daily quests completed and

w

represent the number of weekly challenges completed. The system is:

\begin{aligned} d + w &= 24 \\ 5d + 15w &= 190 \end{aligned}

This sets up the mathematical model using variables for the two unknown quantities.

2

Express one variable in terms of the other using the first equation.

d = 24 - w

This allows for substitution into the second equation to reduce the system to a single variable.

3

Substitute the expression into the second equation and expand.

5(24 - w) + 15w = 190 \implies 120 - 5w + 15w = 190

This substitutes the representation of

d

to solve for

w

directly.

4

Simplify the equation and solve for

w

.

120 + 10w = 190 \implies 10w = 70 \implies w = 7

Combining like terms and isolating the variable yields the number of weekly challenges.

Key Concept

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

Alternative Method

Instead of substitution, the elimination method can be used. Multiply the first equation,

d + w = 24

, by

-5

to get

-5d - 5w = -120

. Add this equation to the second equation,

5d + 15w = 190

, to eliminate

d

, resulting in

10w = 70

, which simplifies to

w = 7

.

Estimated Time:1m 30s

Question 350Question →

A landscaping company sells two types of soil mixtures: a basic mixture and a premium mixture. Each bag of basic mixture contains $4$ pounds of compost and $8$ pounds of sand. Each bag of premium mixture contains $6$ pounds of compost and $5$ pounds of sand. A landscaper purchases a combination of bags containing a total of $46$ pounds of compost and $57$ pounds of sand. How many bags of premium mixture did the landscaper purchase?

Show answer & explanation

Answer: 5

Answer

5

The system of equations representing the scenario is

4b + 6p = 46

for compost and

8b + 5p = 57

for sand, where

b

represents the number of bags of basic mixture and

p

represents the number of bags of premium mixture. Multiplying the first equation by

2

gives

8b + 12p = 92

. Subtracting the second equation from this yields

(8b + 12p) - (8b + 5p) = 92 - 57

, which simplifies to

7p = 35

. Dividing by

7

gives

p = 5

. Thus, the landscaper purchased

5

bags of premium mixture.

Step-by-Step Solution

1

Define variables and write the system of equations based on the given context.

Let

b

represent the number of basic mixture bags and

p

represent the number of premium mixture bags. The system is:

\begin{aligned} 4b + 6p &= 46 \\ 8b + 5p &= 57 \end{aligned}

Translating the word problem into a system of linear equations is necessary to solve for the unknowns.

2

Multiply the first equation by

2

to facilitate the elimination method.

8b + 12p = 92

Aligning the coefficients of

b

allows us to eliminate

b

by subtracting the two equations.

3

Subtract the second equation from the new equation.

(8b + 12p) - (8b + 5p) = 92 - 57 \implies 7p = 35

This isolates the variable

p

by eliminating the variable

b

.

4

Solve for

p

.

p = 5

Dividing both sides of the equation by

7

gives the final number of premium mixture bags.

Key Concept

Solving systems of linear equations in context

Question 351Question →

A commercial bakery uses a mixing bowl that initially contains some flour. A machine adds flour to the bowl at a constant rate. After the machine has been running for $8$ minutes, the total mass of the flour in the bowl is $14$ kilograms. After the machine has been running for $20$ minutes, the total mass of the flour in the bowl is $29$ kilograms. The relationship between the total mass of the flour in the bowl, $M$ , in kilograms, and the time the machine has been running, $t$ , in minutes, is linear. What is the mass of the flour, in kilograms, in the bowl before the machine starts running?

Show answer & explanation

Answer: 4

Answer

4

The relationship between the total mass of the flour,

M

, and the time,

t

, is linear and can be represented by the equation

M = mt + b

, where

m

is the rate at which flour is added and

b

is the initial mass of the flour in the bowl. Using the two given points,

(8, 14)

and

(20, 29)

, the slope

m

is calculated as

m = \frac{29 - 14}{20 - 8} = \frac{15}{12} = 1.25

kilograms per minute. Substituting

m = 1.25

and the point

(8, 14)

into the linear equation gives

14 = 1.25(8) + b

, which simplifies to

14 = 10 + b

. Solving for

b

yields

b = 4

. Therefore, the mass of the flour in the bowl before the machine starts running is

4

kilograms.

Step-by-Step Solution

1

Find the rate of change (slope) of the linear relationship.

The rate is

1.25

kilograms per minute.

The slope of a linear relationship represents the constant rate at which flour is added to the bowl.

2

Set up the linear equation and solve for the y-intercept.

The initial mass is

4

kilograms.

The y-intercept represents the initial mass of the flour in the bowl at

t = 0

minutes.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 352Question →

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(k, 2k + 3)$ and $(2k, 5k - 1)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $f$ is $-9$ , what is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of

k

is 16.

The value of

k

is 16 because when

k = 16

, the points on the graph are

(16, 35)

and

(32, 79)

. The slope of the line is

\frac{79 - 35}{32 - 16} = 2.75

. The equation of the line in slope-intercept form is

y = 2.75x + b

. Using the point

(16, 35)

, we get

35 = 2.75(16) + b \implies 35 = 44 + b \implies b = -9

, which matches the given

y

-intercept of

-9

.

Step-by-Step Solution

1

Calculate the slope of the line in terms of

k

using the two given points

(k, 2k + 3)

and

(2k, 5k - 1)

.

The slope

m

is

\frac{3k - 4}{k}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Use the

y

-intercept of

-9

, which corresponds to the point

(0, -9)

, along with the point

(k, 2k + 3)

to write another expression for the slope.

The slope

m

is

\frac{2k + 12}{k}

.

The slope must be constant for all points on the line, so the slope between the

y

-intercept and one of the points must equal the slope between the two points.

3

Equate the two slope expressions and solve for

k

.

k = 16

Setting the two expressions for the slope equal to each other gives

\frac{3k - 4}{k} = \frac{2k + 12}{k}

. Multiplying by

k

on both sides yields

3k - 4 = 2k + 12

, which simplifies to

k = 16

.

Key Concept

Linear Functions and Graphs

Question 353Question →

In the $xy$ -plane, a line passes through the point $(5, -1)$ and has a slope of $\frac{2}{5}$ . If the line also passes through the point $(15, p)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 3

Answer

The value of

p

is

3

.

To find the value of

p

, the equation of the line can be established using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the given slope

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this yields

y + 1 = \frac{2}{5}x - 2

, which reduces to

y = \frac{2}{5}x - 3

. Substituting the point

(15, p)

into this equation gives

p = \frac{2}{5}(15) - 3 = 6 - 3 = 3

.

Step-by-Step Solution

1

Determine the equation of the line using point-slope form.

y = \frac{2}{5}x - 3

The equation of a line with slope

m

passing through a point

(x_1, y_1)

is

y - y_1 = m(x - x_1)

. Substituting

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this equation results in

y + 1 = \frac{2}{5}x - 2

, which becomes

y = \frac{2}{5}x - 3

.

2

Substitute the point

(15, p)

into the linear equation.

p = 3

Since the line passes through the point

(15, p)

, the coordinates must satisfy the equation of the line. Substituting

x = 15

and

y = p

into

y = \frac{2}{5}x - 3

gives

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

, which simplifies to

p = 6 - 3 = 3

.

Key Concept

Using the slope and a point on a line to find another coordinate along the same line.

Question 354Question →

A linear relationship between $x$ and $y$ is represented by the values in the table below.

$x$	$y$
$-2$	$a$
$1$	$5$
$4$	$17$
$7$	$b$

What is the value of $b - a$ ?

2.25

9

36

22

Show answer & explanation

Answer: 36

Answer

36

The constant rate of change (slope) of the relationship is 4, which is found by dividing the difference in

y

-values by the difference in

x

-values for the given points:

\frac{17 - 5}{4 - 1} = 4

. The value of

b - a

represents the change in

y

over the interval from

x = -2

to

x = 7

. The length of this interval is

7 - (-2) = 9

. Multiplying the rate of change by this interval length gives the total change in

y

:

4 \times 9 = 36

. Alternatively, solving for the equation yields

y = 4x + 1

, where substituting

x = -2

gives

a = -7

and substituting

x = 7

gives

b = 29

, and their difference is

29 - (-7) = 36

.

Step-by-Step Solution

1

Find the constant rate of change (slope) of the linear relationship using the points

(1, 5)

and

(4, 17)

.

The slope

m

is

\frac{17 - 5}{4 - 1} = \frac{12}{3} = 4

.

Since the relationship is linear, the rate of change is constant between any two points.

2

Determine the value of

a

when

x = -2

.

Using the point

(1, 5)

and moving to

x = -2

, the change in

x

is

-3

. The corresponding change in

y

is

4 \times (-3) = -12

. Thus,

a = 5 - 12 = -7

.

This establishes the value of the first variable in the expression.

3

Determine the value of

b

when

x = 7

.

Using the point

(4, 17)

and moving to

x = 7

, the change in

x

is

+3

. The corresponding change in

y

is

4 \times 3 = 12

. Thus,

b = 17 + 12 = 29

.

This establishes the value of the second variable in the expression.

4

Calculate the value of

b - a

.

b - a = 29 - (-7) = 29 + 7 = 36

.

Subtracting a negative number is equivalent to adding its positive counterpart.

Key Concept

Linear rate of change and evaluation of linear relationships from tables of values

Alternative Method

Instead of calculating the individual values of

a

and

b

, recognize that

b - a

is the total change in

y

over the interval from

x = -2

to

x = 7

. The change in

x

is

7 - (-2) = 9

. Since the constant rate of change (slope) is

4

, the change in

y

is simply

4 \times 9 = 36

.

Estimated Time:1m 30s

Question 355Question →

\begin{aligned} 3x - 2y &= 14 \\ 5x + 6y &= 42 \end{aligned}

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

2

4

5

8

Show answer & explanation

Answer: 4

Answer

The value of

x - y

is

4

.

To find the value of

x - y

, we first solve the system of linear equations. Multiplying the first equation,

3x - 2y = 14

, by

3

gives

9x - 6y = 42

. Adding this equation to the second equation,

5x + 6y = 42

, eliminates

y

and yields

14x = 84

, which simplifies to

x = 6

. Substituting

x = 6

back into the first equation gives

3(6) - 2y = 14

, or

18 - 2y = 14

, which simplifies to

2y = 4

, so

y = 2

. Therefore, the value of

x - y

is

6 - 2 = 4

.

Step-by-Step Solution

1

Multiply the first equation by

3

to align the coefficients of the

y

terms for elimination.

9x - 6y = 42

This makes the coefficients of

y

opposite in sign and equal in magnitude to the second equation.

2

Add the modified first equation to the second equation to eliminate

y

and solve for

x

.

(9x - 6y) + (5x + 6y) = 42 + 42 \implies 14x = 84 \implies x = 6

Adding the equations eliminates

y

, resulting in a single-variable linear equation.

3

Substitute

x = 6

back into the first equation to solve for

y

.

3(6) - 2y = 14 \implies 18 - 2y = 14 \implies -2y = -4 \implies y = 2

Substituting the known value of

x

yields the value of the other coordinate.

4

Evaluate the expression

x - y

using the values

x = 6

and

y = 2

.

x - y = 6 - 2 = 4

This computes the requested quantity.

Key Concept

Solving systems of linear equations using the elimination method and evaluating a linear combination of the variables.

Question 356Question →

A municipal recycling facility processes plastic waste. The total mass of unprocessed plastic, $M$ , in tons, remaining at the facility $h$ hours after the facility opens on a given day can be modeled by the equation $M = 142 - 8.5h$ . Which of the following is the best interpretation of the number $8.5$ in this context?

The facility has

142

tons of unprocessed plastic when it opens.

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

The facility takes

8.5

hours to process each ton of plastic.

The mass of unprocessed plastic at the facility decreases by

133.5

tons each hour.

Show answer & explanation

Answer: The mass of unprocessed plastic at the facility decreases by $8.5$ tons each hour.

Answer

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

In the equation

M = 142 - 8.5h

, the total mass of unprocessed plastic

M

decreases by

8.5

tons for each hour

h

that passes. The number

8.5

is the magnitude of the slope of the linear relationship, which represents the rate of change. Since the coefficient of

h

is negative, the amount of unprocessed plastic decreases by

8.5

tons per hour.

Step-by-Step Solution

1

Identify the structure of the linear equation

M = 142 - 8.5h

.

The equation is in the form

y = b + mx

, where

b

is the y-intercept (

142

) and

m

is the slope (

-8.5

).

This helps separate the initial value from the rate of change.

2

Analyze the coefficient of the independent variable

h

.

The coefficient is

-8.5

.

The coefficient of

h

represents the rate of change of the mass

M

per hour.

3

Interpret the meaning of the rate of change in context.

A rate of change of

-8.5

means that the mass of unprocessed plastic decreases by

8.5

tons for each hour that passes.

The negative sign indicates a decrease, and the unit is tons per hour.

Key Concept

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

Question 357Question →

\begin{aligned} 5x + 3y &= 22 \\ 3x + 5y &= 18 \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: 2

Answer

The value of the expression

x - y

is 2.

Subtracting the second equation,

3x + 5y = 18

, from the first equation,

5x + 3y = 22

, yields

(5x - 3x) + (3y - 5y) = 22 - 18

, which simplifies to

2x - 2y = 4

. Dividing both sides of this equation by 2 isolates the expression

x - y

and yields a final value of 2.

Step-by-Step Solution

1

Subtract the second equation from the first equation.

2x - 2y = 4

Subtracting the equations aligns the coefficients of

x

and

y

to form a multiple of the target expression

x - y

.

2

Divide both sides of the equation by 2.

x - y = 2

Dividing the expression

2x - 2y

by 2 isolates the target expression

x - y

.

Key Concept

Solving systems of linear equations by linear combination and algebraic manipulation.

Question 358Question →

A researcher uses the linear equation $T = 18.5 + 4.2d$ to estimate the temperature, $T$ , in degrees Celsius, of the Earth's crust at a depth of $d$ kilometers below the surface in a certain region. What is the estimated increase in temperature, in degrees Celsius, for each increase of 5 kilometers in depth?

Show answer & explanation

Answer: 21

Answer

21

The coefficient of

d

in the equation is

4.2

, representing a temperature increase of

4.2

degrees Celsius for every 1 kilometer increase in depth. To find the temperature increase for an increase of 5 kilometers in depth, multiply this rate of change by 5:

4.2 \times 5 = 21

.

Step-by-Step Solution

1

Identify the slope of the linear equation.

The slope is

4.2

.

In the equation

T = 18.5 + 4.2d

, the term

4.2d

indicates that for every 1 kilometer increase in depth (

d

), the temperature (

T

) increases by

4.2

degrees Celsius.

2

Calculate the total temperature increase for a 5-kilometer depth increase.

The increase is

21

degrees Celsius.

Since the rate of temperature increase is

4.2

degrees Celsius per kilometer, a depth increase of

5

kilometers results in a temperature increase of

4.2 \times 5 = 21

degrees Celsius.

Key Concept

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

Estimated Time:1m 30s

Question 359Question →

A commercial printing press uses a continuous roll of paper to print newspapers at a constant rate. After $10$ minutes of operation, the remaining length of the paper roll is $12,500$ feet. After $25$ minutes of operation, the remaining length of the paper roll is $8,000$ feet. If the relationship between the printing time, in minutes, and the remaining length of the paper roll, in feet, is linear, how many minutes after the printing press starts operating will the remaining length of the paper roll be $2,000$ feet?

Show answer & explanation

Answer: 45

Answer

The remaining length of the paper roll will be

2,000

feet after

45

minutes of operation.

We are given that the remaining length of the paper roll is a linear function of time,

t

. Let

L(t)

be the remaining length of the paper roll, in feet, after

t

minutes of operation. We can represent the given information as two coordinate points:

(10, 12500)

and

(25, 8000)

. First, find the slope, which represents the constant rate at which the paper is consumed:

m = \frac{8000 - 12500}{25 - 10} = \frac{-4500}{15} = -300

feet per minute. Next, write the linear equation using the point-slope form:

L(t) - 12500 = -300(t - 10)

, which simplifies to

L(t) = 15500 - 300t

. To find the time when the remaining length is

2,000

feet, set

L(t) = 2000

and solve for

t

:

2000 = 15500 - 300t

, which simplifies to

300t = 13500

, giving

t = 45

.

Step-by-Step Solution

1

Calculate the rate of paper consumption (the slope of the linear equation) using the two given points,

(10, 12500)

and

(25, 8000)

.

The rate of paper consumption is

-300

feet per minute.

To establish the linear relationship, we first need the constant rate of change (slope) from the two known coordinate points.

2

Use the point-slope equation of a line,

y - y_1 = m(x - x_1)

, with the point

(10, 12500)

and slope

m = -300

, to find the equation relating the remaining length,

L

, to the time,

t

.

L(t) = 15500 - 300t

We need the full linear model to calculate the remaining length at any specific time.

3

Substitute

L(t) = 2000

into the linear equation and solve for

t

.

t = 45

This gives the specific operating time in minutes when the remaining paper roll length is

2,000

feet.

Key Concept

Writing and solving linear equations in two variables from two coordinate points.

Question 360Question →

A software developer uses the equation $P = 1500 + 45n$ to calculate the total price $P$ , in dollars, charged to a client for a project, where $n$ represents the number of hours spent debugging the software. Which of the following is the best interpretation of the number $45$ in this context?

The increase in the total price, in dollars, for each additional hour spent debugging the software

The initial price, in dollars, charged for the project before any hours are spent debugging

The number of additional hours spent debugging for each 1-dollar increase in the total price

The total price, in dollars, charged for a project that requires exactly 1 hour of debugging

Show answer & explanation

Answer: The increase in the total price, in dollars, for each additional hour spent debugging the software

Answer

The increase in the total price, in dollars, for each additional hour spent debugging the software

The coefficient of

n

in the equation

P = 1500 + 45n

is

45

, which represents the slope of the line. The slope measures the rate of change of the dependent variable,

P

(total price in dollars), per unit change in the independent variable,

n

(hours spent debugging). Thus,

45

represents an increase of 45 dollars in the total price for each additional hour spent debugging.

Step-by-Step Solution

1

Identify the structure of the linear equation.

The equation

P = 1500 + 45n

is in the slope-intercept form

y = mx + b

, where

P

is the dependent variable (equivalent to

y

),

n

is the independent variable (equivalent to

x

), the coefficient

45

is the slope

m

, and the constant

1500

is the y-intercept

b

.

Understanding the components of a linear equation helps map them to their real-world meanings.

2

Interpret the meaning of the slope in context.

The slope

m = 45

represents the rate of change of the dependent variable

P

(total price in dollars) with respect to the independent variable

n

(hours spent debugging). This means that for each unit increase in

n

(1 additional hour of debugging),

P

increases by

45

units (45 dollars).

The coefficient of the independent variable in a linear model represents the unit rate of change.

Key Concept

Interpreting the slope of a linear model in context

Algebra

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