Linear Equations in Two Variables Practice Questions — SAT — Page 3

Question 41Question →

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 42Question →

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 43Question →

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 44Question →

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 45Question →

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 46Question →

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 47Question →

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 48Question →

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 49Question →

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 50Question →

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 51Question →

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 52Question →

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.

Question 53Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the points $(-2, 5)$ and $(4, 7)$ . What is the value of $m + b$ ?

6

-2

4

\frac{14}{3}

Show answer & explanation

Answer: $6$

Answer

The value of

m + b

is

6

.

The correct answer is

6

. By finding the slope

m

using the change in

y

divided by the change in

x

, we get

m = \frac{1}{3}

. Using the slope-intercept form with the point

(4, 7)

allows us to solve for

b = \frac{17}{3}

. Adding the two values together yields

m + b = \frac{1}{3} + \frac{17}{3} = 6

.

Step-by-Step Solution

1

Calculate the slope

m

of the line using the two points

(-2, 5)

and

(4, 7)

.

m = \frac{7 - 5}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}

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

Substitute the slope

m = \frac{1}{3}

and the coordinates of one point, such as

(4, 7)

, into the slope-intercept equation

y = mx + b

to solve for

b

.

7 = \frac{1}{3}(4) + b \Rightarrow 7 = \frac{4}{3} + b \Rightarrow b = \frac{17}{3}

Substituting a known point and the slope into the slope-intercept form allows us to isolate and solve for the constant

y

-intercept.

3

Add the calculated values of

m

and

b

to find

m + b

.

m + b = \frac{1}{3} + \frac{17}{3} = \frac{18}{3} = 6

The question asks for the sum of the slope

m

and the

y

-intercept

b

.

Key Concept

Linear Equations in Two Variables

Question 54Question →

In the $xy$ -plane, a line is represented by the equation $ay - bx = 24$ , where $a$ and $b$ are constants. If the line has a $y$ -intercept of $(0, -3)$ and passes through the point $(4, 5)$ , what is the slope of the line?

Show answer & explanation

Answer: 2

Answer

The slope of the line is 2.

Substituting the y-intercept

(0, -3)

into the equation

ay - bx = 24

yields

-3a = 24

, which gives

a = -8

. Substituting the point

(4, 5)

and

a = -8

into the equation yields

-8(5) - 4b = 24

, which simplifies to

-40 - 4b = 24

, giving

b = -16

. Re-assembling the equation gives

-8y + 16x = 24

. Solving for

y

in terms of

x

yields

y = 2x - 3

, where the coefficient of

x

is the slope, 2.

Step-by-Step Solution

1

Substitute the y-intercept

(0, -3)

into the given equation

ay - bx = 24

to find the value of

a

.

a = -8

Since the y-intercept lies on the line, its coordinates satisfy the line's equation.

2

Substitute the point

(4, 5)

and the value of

a = -8

into the equation to find the value of

b

.

b = -16

Since the point

(4, 5)

lies on the line, its coordinates must satisfy the equation.

3

Write the resulting equation

-8y + 16x = 24

in slope-intercept form (

y = mx + c

) to identify the slope.

y = 2x - 3

, which gives a slope of

2

.

The slope of a line in the form

y = mx + c

is represented by the coefficient

m

of

x

.

Key Concept

Finding the slope of a line from a given equation by determining its constant coefficients using known points.

Estimated Time:1m 30s

Question 55Question →

An online retail company determines that the relationship between the selling price of a product, $x$ dollars, and the daily number of units sold, $y$ , can be modeled by a linear equation. When the selling price is $12$ dollars, the company sells $80$ units per day. For every $3$ dollars increase in the selling price, the number of units sold daily decreases by $15$ . Which of the following equations represents this relationship?

y = -5x + 140

y = -5x + 80

y = -0.2x + 82.4

y = -5x + 20

Show answer & explanation

Answer: $y = -5x + 140$

Answer

The equation

y = -5x + 140

The relationship between price

x

and units sold

y

is linear. The slope

m

represents the rate of change:

m = \frac{\Delta y}{\Delta x} = \frac{-15}{3} = -5

. Using the point-slope form with the known point

(12, 80)

, we get

y - 80 = -5(x - 12)

. Distributing the

-5

gives

y - 80 = -5x + 60

. Adding

80

to both sides results in the equation

y = -5x + 140

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear relation.

m = \frac{\text{change in } y}{\text{change in } x} = \frac{-15}{3} = -5

The slope is the rate of change, which represents the decrease of

15

units for every

3

dollars increase in price.

2

Set up the equation using point-slope form with the point

(12, 80)

and the slope

m = -5

.

y - 80 = -5(x - 12)

Point-slope form allows us to write the equation of a line given its slope and a point it passes through.

3

Simplify the equation into slope-intercept form.

y - 80 = -5x + 60 \Rightarrow y = -5x + 140

Distributing the slope

-5

to

-12

yields

+60

, and adding

80

to both sides isolates the variable

y

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Substitute the point

(12, 80)

and slope

m = -5

into the slope-intercept equation

y = mx + b

to find

b

:

80 = -5(12) + b \Rightarrow 80 = -60 + b \Rightarrow b = 140

. Thus,

y = -5x + 140

.

Estimated Time:1m 30s

Question 56Question →

The graph of a linear equation in the $xy$ -plane has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, 3k)$ , where $k$ is a positive constant. If the line passes through the point $(2, 12)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

The value of

k

is

6

.

The correct answer is

6

. The slope of the line can be found using the two intercepts

(k, 0)

and

(0, 3k)

:

m = \frac{3k - 0}{0 - k} = -3

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

y = -3x + 3k

. Since the line passes through the point

(2, 12)

, we substitute

x = 2

and

y = 12

into the equation to get

12 = -3(2) + 3k

. Simplifying the equation yields

12 = -6 + 3k

, which gives

18 = 3k

. Dividing both sides by

3

results in

k = 6

.

Step-by-Step Solution

1

Find the slope of the line using the

x

-intercept

(k, 0)

and the

y

-intercept

(0, 3k)

.

The slope of the line is

-3

.

The slope formula is

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

. Substituting the points

(k, 0)

and

(0, 3k)

gives

m = \frac{3k - 0}{0 - k} = \frac{3k}{-k} = -3

.

2

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

The equation of the line is

y = -3x + 3k

.

Using the slope-intercept form

y = mx + b

, we substitute the slope

m = -3

and the

y

-intercept value

b = 3k

from the point

(0, 3k)

.

3

Substitute the given point

(2, 12)

into the equation and solve for

k

.

The value of

k

is

6

.

Substituting

x = 2

and

y = 12

into

y = -3x + 3k

yields

12 = -3(2) + 3k

. This simplifies to

12 = -6 + 3k

. Adding

6

to both sides gives

18 = 3k

, and dividing by

3

gives

k = 6

.

Key Concept

Linear equations in two variables, finding equations from intercepts, and constant determination via point substitution.

Question 57Question →

A municipal swimming pool is being filled with water at a constant rate. After $2$ hours of filling, the pool contains $14,500$ gallons of water. After $5$ hours of filling, the pool contains $20,200$ gallons of water. If the relationship between the time the pool has been filling, $t$ , in hours, and the volume of water in the pool, $V$ , in gallons, is linear, which of the following equations represents this relationship?

V = 1,900t + 14,500

V = 5,700t + 3,100

V = 1,900t + 10,700

V = 1,900t + 18,300

Show answer & explanation

Answer: $V = 1,900t + 10,700$

Answer

The equation representing the relationship is

V = 1,900t + 10,700

.

The correct equation is

V = 1,900t + 10,700

. The rate of change of the water volume is the change in volume divided by the change in time:

\frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900

gallons per hour. Using the point-slope form with the coordinate point

(2, 14,500)

yields

V - 14,500 = 1,900(t - 2)

. Simplifying this expression gives

V - 14,500 = 1,900t - 3,800

, which results in

V = 1,900t + 10,700

.

Step-by-Step Solution

1

Identify two data points from the problem context.

The two coordinate pairs representing

(t, V)

are

(2, 14,500)

and

(5, 20,200)

.

These points will allow us to calculate the slope and the y-intercept of the linear equation.

2

Calculate the slope (

m

) using the slope formula

m = \frac{V_2 - V_1}{t_2 - t_1}

.

m = \frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900

.

The slope represents the constant rate, in gallons per hour, at which the pool is being filled.

3

Substitute the slope

m = 1,900

and the point

(2, 14,500)

into the point-slope form equation

V - V_1 = m(t - t_1)

to solve for

V

.

V - 14,500 = 1,900(t - 2) \implies V - 14,500 = 1,900t - 3,800 \implies V = 1,900t + 10,700

.

This yields the equation representing the volume of water

V

in the pool at any time

t

.

Key Concept

Determining a linear equation in two variables given two points from a word problem context.

Alternative Method

Instead of solving the linear equation algebraically, you can test the coordinates of the two given points

(2, 14,500)

and

(5, 20,200)

in the answer choices. Substituting

t = 2

and

t = 5

into the correct equation

V = 1,900t + 10,700

satisfies both conditions:

1,900(2) + 10,700 = 14,500

and

1,900(5) + 10,700 = 20,200

. None of the other options satisfy both points.

Estimated Time:1m 30s

Question 58Question →

A technician is monitoring the pressure of a gas inside a container during an experiment. The pressure $P$ , in kilopascals ( $\text{kPa}$ ), and the time elapsed $t$ , in minutes, are related by a linear equation. At $t = 4$ minutes, the pressure is $112\text{ kPa}$ . At $t = 12$ minutes, the pressure is $136\text{ kPa}$ . If the pressure continues to increase at this constant rate, what will the pressure be, in $\text{kPa}$ , at $t = 15$ minutes?

Show answer & explanation

Answer: 145

Answer

145

The relationship between pressure

P

and time

t

is linear, which can be modeled by the equation

P = mt + b

, where

m

is the rate of change (slope) and

b

is the pressure at

t = 0

. Using the points

(4, 112)

and

(12, 136)

, the slope is calculated as

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

. Substituting the point

(4, 112)

and

m = 3

into the equation

P = mt + b

yields

112 = 3(4) + b

, which simplifies to

112 = 12 + b

, so

b = 100

. The linear equation is

P = 3t + 100

. Substituting

t = 15

into this equation gives

P = 3(15) + 100 = 45 + 100 = 145

.

Step-by-Step Solution

1

Calculate the rate of change (slope,

m

) using the two given coordinate points

(4, 112)

and

(12, 136)

.

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

To find the constant rate at which the pressure is increasing per minute.

2

Set up the linear equation using the point-slope form

P - P_1 = m(t - t_1)

with the point

(4, 112)

.

P - 112 = 3(t - 4) \Rightarrow P = 3t + 100

To establish the linear relationship between pressure and time.

3

Substitute

t = 15

into the linear equation to find the pressure at

15

minutes.

P = 3(15) + 100 = 145

To determine the pressure at the requested time of

15

minutes.

Key Concept

Finding and applying a linear equation in two variables from two points.

Question 59Question →

The total daily cost $C$ , in dollars, for a custom apparel company to manufacture $x$ shirts is modeled by a linear equation. The table below shows the total daily cost for two different numbers of shirts manufactured:

Number of shirts, $x$	Total daily cost, $C$ (dollars)
12	260
20	380

Based on the model, what is the total daily cost, in dollars, to manufacture 35 shirts?

262

536

605

785

Show answer & explanation

Answer: 605

Answer

605

To find the total daily cost to manufacture 35 shirts, we first determine the linear relationship

C = mx + b

between the number of shirts manufactured,

x

, and the total cost,

C

. The slope

m

is the constant rate of change, calculated as

m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15

dollars per shirt. Using the point-slope form with the point

(12, 260)

, we get

C - 260 = 15(x - 12)

, which simplifies to

C = 15x + 80

. Substituting

35

for

x

in this equation yields

C = 15(35) + 80 = 525 + 80 = 605

. Therefore, the total daily cost to manufacture 35 shirts is 605 dollars.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear relationship using the two points from the table,

(12, 260)

and

(20, 380)

.

Slope

m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15

dollars per shirt.

Since the relationship is linear, the rate of change is constant.

2

Find the equation of the line using the point-slope form

C - C_1 = m(x - x_1)

with the point

(12, 260)

and slope

m = 15

.

C - 260 = 15(x - 12) \implies C = 15x - 180 + 260 \implies C = 15x + 80

.

This establishes the linear equation relating the number of shirts

x

and the total daily cost

C

.

3

Substitute

x = 35

into the linear equation to find the total daily cost to manufacture 35 shirts.

C = 15(35) + 80 = 525 + 80 = 605

dollars.

This evaluates the linear model at the desired quantity.

Key Concept

Determining a linear equation in two variables from a table of values and using it to predict a value.

Question 60Question →

A hybrid car's fuel tank has a capacity of $12$ gallons. When driving on a highway, the amount of fuel in the tank decreases at a constant rate. After driving for $1.5$ hours, $9.6$ gallons of fuel remain in the tank. If the amount of fuel in the tank, $F$ , in gallons, after driving for $t$ hours is modeled by a linear equation, what is the value of $F$ when $t = 4$ ?

Show answer & explanation

Answer: 5.6

Answer

The amount of fuel remaining in the tank after driving for

4

hours is

5.6

gallons.

The amount of fuel in the tank,

F

, and the driving time,

t

, share a linear relationship. The initial amount of fuel at

t = 0

is

12

gallons, representing the vertical intercept. The fuel decreases at a constant rate, which is the slope of the line. Over a period of

1.5

hours, the amount of fuel decreases by

12 - 9.6 = 2.4

gallons. The rate of decrease is

\frac{2.4}{1.5} = 1.6

gallons per hour, so the slope is

-1.6

. The linear model is

F = -1.6t + 12

. Substituting

t = 4

into this equation yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

gallons.

Step-by-Step Solution

1

Calculate the constant rate of fuel consumption (the slope of the linear relationship).

The rate of consumption is

1.6

gallons per hour.

Since the fuel decreases at a constant rate, the change in fuel divided by the change in time gives the rate of consumption. In

1.5

hours, the fuel decreases from

12

gallons to

9.6

gallons, which is a decrease of

12 - 9.6 = 2.4

gallons. Thus, the rate of consumption is

\frac{2.4\text{ gallons}}{1.5\text{ hours}} = 1.6

gallons per hour.

2

Write the linear equation modeling the fuel remaining in the tank,

F

, as a function of time,

t

.

F = -1.6t + 12

The initial amount of fuel when

t = 0

is

12

gallons, which represents the vertical intercept (

b = 12

). The fuel decreases at a constant rate of

1.6

gallons per hour, which represents a slope of

m = -1.6

.

3

Substitute

t = 4

into the linear equation to find the value of

F

.

F = 5.6

Evaluating the equation at

t = 4

yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

.

Key Concept

Linear Equations in Two Variables

Linear Equations in Two Variables

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes