Linear Equations in Two Variables Practice Questions — SAT — Page 2

Question 21Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the point $(2, 7)$ . If this graph is translated $3$ units to the right and $4$ units down, the resulting graph passes through the point $(4, 2)$ . What is the value of $b$ ?

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. Since the original graph passes through

(2, 7)

and undergoes a translation of

3

units to the right and

4

units down, the point

(2, 7)

is translated to

(2 + 3, 7 - 4) = (5, 3)

on the new graph. The translated graph also passes through

(4, 2)

. Using the slope formula on the two points

(5, 3)

and

(4, 2)

of the translated line gives a slope of

1

. Since a translation preserves the slope, the original line also has a slope of

1

. Substituting

m = 1

and

(2, 7)

into the original equation

y = mx + b

yields

7 = 1(2) + b

, which simplifies to

b = 5

.

Step-by-Step Solution

1

Determine a point on the translated graph by translating the given point

(2, 7)

on the original graph.

The point

(5, 3)

is on the translated graph.

Since the entire graph is translated

3

units to the right and

4

units down, every point on the original graph is translated by the same vector

(+3, -4)

.

2

Calculate the slope of the translated graph using the points

(5, 3)

and

(4, 2)

.

The slope

m = 1

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the change in y divided by the change in x.

3

Find the slope of the original graph.

The slope of the original graph is also

1

.

Translating a line horizontally and vertically shifts its position but does not alter its slope.

4

Substitute the slope

m = 1

and the point

(2, 7)

into the slope-intercept form of the original line,

y = mx + b

, to solve for

b

.

b = 5

.

This determines the value of the y-intercept of the original line.

Key Concept

Linear Equations in Two Variables

Question 22Question →

A linear relationship exists between the variables $p$ and $q$ . When the value of $p$ increases by $6$ , the value of $q$ decreases by $4$ . If $q = 15$ when $p = 2$ , which of the following equations correctly expresses $p$ in terms of $q$ ?

p = -\frac{3}{2}q + \frac{49}{2}

p = -\frac{2}{3}q + \frac{49}{3}

p = \frac{3}{2}q - \frac{41}{2}

p = -\frac{2}{3}q + 12

Show answer & explanation

Answer: $p = -\frac{3}{2}q + \frac{49}{2}$

Answer

p = -\frac{3}{2}q + \frac{49}{2}

The correct equation is found by first calculating the rate of change of

q

with respect to

p

, which is

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

. Using the point-slope form with

(p, q) = (2, 15)

, we set up the equation

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

. To express

p

in terms of

q

, we isolate

p

by multiplying both sides by

-\frac{3}{2}

to get

-\frac{3}{2}q + \frac{45}{2} = p - 2

, and then adding

2

to both sides, which yields

p = -\frac{3}{2}q + \frac{49}{2}

.

Step-by-Step Solution

1

Calculate the rate of change of

q

relative to

p

using the given changes.

The rate of change is

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

.

Since

q

decreases by

4

when

p

increases by

6

, the slope

m

in terms of

\Delta q / \Delta p

is

-\frac{2}{3}

.

2

Set up the point-slope form of the linear equation using the point

(p, q) = (2, 15)

.

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

The point-slope form is

q - q_1 = m(p - p_1)

, where

(p_1, q_1) = (2, 15)

and

m = -\frac{2}{3}

.

3

Solve the equation for

p

by first clearing the coefficient of the

p

-term.

-\frac{3}{2}(q - 15) = p - 2 \implies -\frac{3}{2}q + \frac{45}{2} = p - 2

Multiplying both sides by

-\frac{3}{2}

simplifies isolation of the variable

p

.

4

Complete the isolation of

p

by adding

2

to both sides.

p = -\frac{3}{2}q + \frac{45}{2} + 2 \implies p = -\frac{3}{2}q + \frac{49}{2}

Adding

2

(which is

\frac{4}{2}

) to

\frac{45}{2}

isolates

p

on one side of the equation.

Key Concept

Linear relationships and isolating variables in two-variable linear equations.

Question 23Question →

In the $xy$ -plane, the graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line. The line has an $x$ -intercept of $(d, 0)$ and a $y$ -intercept of $(0, d - 6)$ , where $d > 6$ is a constant. If the line passes through the point $(2, 5)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 6

Answer

6

Substituting the intercepts

(d, 0)

and

(0, d-6)

into

ax + by = 24

gives

a = 24/d

and

b = 24/(d-6)

. Using the point

(2, 5)

, we get

2(24/d) + 5(24/(d-6)) = 24

, which simplifies to

2/d + 5/(d-6) = 1

. Solving for

d

yields

d^2 - 13d + 12 = 0

, giving

d = 12

or

d = 1

. Given

d > 6

, we have

d = 12

. Substituting

d = 12

back gives

a = 2

and

b = 4

, so

a + b = 6

.

Step-by-Step Solution

1

Find expressions for

a

and

b

in terms of

d

by substituting the given intercepts.

a = \frac{24}{d}

and

b = \frac{24}{d-6}

The

x

-intercept

(d, 0)

and

y

-intercept

(0, d-6)

lie on the line

ax + by = 24

.

2

Substitute the point

(2, 5)

into the line's equation.

2a + 5b = 24

The point

(2, 5)

lies on the line.

3

Substitute the expressions for

a

and

b

into

2a + 5b = 24

and simplify.

\frac{2}{d} + \frac{5}{d-6} = 1

To create a single equation in terms of the variable

d

.

4

Solve the rational equation for

d

.

d = 12

Multiplying by the common denominator

d(d-6)

leads to the quadratic equation

d^2 - 13d + 12 = 0

, which factors as

(d-12)(d-1) = 0

. Since the problem states

d > 6

, the only valid solution is

d = 12

.

5

Calculate

a

and

b

using

d = 12

, and find their sum.

a = 2

,

b = 4

, and

a + b = 6

Substituting

d = 12

into the expressions for

a

and

b

yields the constant coefficients, and summing them provides the final requested value.

Key Concept

Using intercepts and a given point on a line in the coordinate plane to solve for the parameters of its linear equation.

Question 24Question →

In the $xy$ -plane, a line with a positive slope $m$ passes through the points $(2, 5)$ and $(m, 13)$ . Which of the following is the $y$ -intercept of this line?

(0, 3)

(0, 9)

(0, -3)

(0, 1)

Show answer & explanation

Answer: $(0, -3)$

Answer

(0, -3)

The correct answer is the point

(0, -3)

. The slope of the line passing through

(2, 5)

and

(m, 13)

is expressed as

m = \frac{13 - 5}{m - 2} = \frac{8}{m - 2}

. Multiplying by

m - 2

yields the quadratic equation

m^2 - 2m - 8 = 0

. Factoring this equation gives

(m - 4)(m + 2) = 0

. Since the slope

m

is given as positive, we select

m = 4

. Using the point-slope form with

m = 4

and the point

(2, 5)

yields the equation

y - 5 = 4(x - 2)

, which simplifies to

y = 4x - 3

. The

y

-intercept of this line is found by setting

x = 0

, giving

y = -3

, or the coordinate point

(0, -3)

.

Step-by-Step Solution

1

Use the slope formula to express the slope of the line passing through

(2, 5)

and

(m, 13)

in terms of

m

.

m = \frac{13 - 5}{m - 2} \implies m = \frac{8}{m - 2}

The slope of a line is defined as the ratio of the change in

y

to the change in

x

between any two points on the line.

2

Solve the equation

m = \frac{8}{m - 2}

for

m

by converting it into a quadratic equation.

m(m - 2) = 8 \implies m^2 - 2m - 8 = 0 \implies (m - 4)(m + 2) = 0

, which gives

m = 4

or

m = -2

. Since the slope is positive,

m = 4

.

Multiplying both sides by the denominator clears the fraction and allows us to solve for the unknown parameter

m

using factoring.

3

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

m = 4

and the point

(2, 5)

.

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

The point-slope form provides a direct way to write the linear equation using a known point and the calculated slope.

4

Find the

y

-intercept by evaluating the line equation at

x = 0

.

y = 4(0) - 3 \implies y = -3

, which corresponds to the point

(0, -3)

.

The

y

-intercept of any line is the point where the line crosses the

y

-axis, which occurs at

x = 0

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Alternatively, once the slope

m = 4

is found, we can write the equation of the line in slope-intercept form

y = 4x + b

. Substituting the coordinates of the point

(2, 5)

into the equation gives

5 = 4(2) + b \implies 5 = 8 + b

. Solving for

b

gives

b = -3

. Since

b

is the

y

-intercept, the point is

(0, -3)

.

Estimated Time:2m 30s

Question 25Question →

For a constant $k$ , the graph of the linear equation $kx - 4y = 15$ in the $xy$ -plane contains the point $(3, 3)$ . What is the slope of the line?

Show answer & explanation

Answer: 2.25

Answer

2.25

Substituting the coordinates of the point

(3, 3)

into the given equation yields

k(3) - 4(3) = 15

, which simplifies to

3k - 12 = 15

. Adding

12

to both sides gives

3k = 27

, and dividing by

3

gives

k = 9

. Substituting

k = 9

back into the original equation gives

9x - 4y = 15

. To find the slope of the line, the equation can be written in slope-intercept form,

y = mx + b

, by isolating

y

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing all terms by

-4

yields

y = \frac{9}{4}x - \frac{15}{4}

, or

y = 2.25x - 3.75

. Therefore, the slope of the line is

\frac{9}{4}

, or

2.25

.

Step-by-Step Solution

1

Substitute the point

(3, 3)

into the equation

kx - 4y = 15

3k - 12 = 15

Since the graph of the equation contains the point

(3, 3)

, substituting

x = 3

and

y = 3

must satisfy the equation.

2

Solve for the constant

k

k = 9

Adding

12

to both sides of the equation gives

3k = 27

. Dividing both sides by

3

yields

k = 9

.

3

Substitute

k = 9

into the equation and rewrite it in slope-intercept form (

y = mx + b

)

y = \frac{9}{4}x - \frac{15}{4}

Substituting

k = 9

gives

9x - 4y = 15

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing both sides by

-4

isolates

y

.

4

Identify the slope from the slope-intercept form

Slope =

2.25

(or

\frac{9}{4}

)

In the slope-intercept form

y = mx + b

, the coefficient

m

represents the slope of the line.

Key Concept

Determining the slope of a line from a linear equation by first finding an unknown constant using a given point on the line.

Question 26Question →

A delivery truck carries a total of $x$ bags of organic fertilizer and $y$ bags of compost. Each bag of fertilizer weighs $45$ pounds, and each bag of compost weighs $30$ pounds. When the truck is loaded to exactly $80\%$ of its maximum payload capacity, the relationship between $x$ and $y$ can be represented by the equation $3x + 2y = 240$ . What is the truck's maximum payload capacity, in pounds?

2,880

3,000

3,600

4,500

Show answer & explanation

Answer: $4,500$

Answer

The maximum payload capacity of the truck is

4,500

pounds.

The total weight of the bags loaded on the truck is represented by the expression

45x + 30y

pounds. Factoring out

15

from this expression gives

15(3x + 2y)

. We are given that when the truck is loaded to exactly

80\%

of its capacity, the relationship between

x

and

y

is

3x + 2y = 240

. Substituting

240

for

3x + 2y

in the weight expression shows that the weight of the load at

80\%

capacity is

15 \times 240 = 3,600

pounds. To find the maximum capacity, we set up the equation

0.80 \times C = 3,600

, where

C

is the maximum payload capacity. Solving for

C

yields

C = \frac{3,600}{0.80} = 4,500

pounds.

Step-by-Step Solution

1

Write the expression for the total weight of the bags in terms of

x

and

y

.

Total weight

W = 45x + 30y

pounds.

Each of the

x

bags of fertilizer weighs

45

pounds and each of the

y

bags of compost weighs

30

pounds.

2

Relate the total weight expression to the given equation for

80\%

capacity.

45x + 30y = 15(3x + 2y)

Factoring out the greatest common divisor of

45

and

30

, which is

15

, allows us to write the weight expression in terms of the left-hand side of the given equation

3x + 2y = 240

.

3

Calculate the actual weight of the load when the truck is at

80\%

capacity.

W = 15 \times 240 = 3,600

pounds.

Since the relationship at

80\%

capacity is

3x + 2y = 240

, we can substitute

240

for

3x + 2y

in the factored weight expression.

4

Calculate the maximum payload capacity of the truck.

Maximum capacity

C = \frac{3,600}{0.80} = 4,500

pounds.

The weight of

3,600

pounds represents

80\%

of the maximum payload capacity, so dividing the actual weight by

0.80

gives the total capacity.

Key Concept

Translating a real-world scenario into a linear equation and interpreting its coefficients and constants to solve for unknown quantities.

Question 27Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has a slope of $\frac{2}{3}$ and passes through the point $(6, 5)$ . If $a + b = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

15

Rearranging the linear equation

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows the slope is

-\frac{a}{b}

. Setting

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

gives

2b = -3a

, or

b = -1.5a

. Substituting this into

a + b = 5

yields

a - 1.5a = 5

, which simplifies to

-0.5a = 5

, so

a = -10

. This means

b = -1.5(-10) = 15

. Substituting the values of

a = -10

and

b = 15

along with the point

(6, 5)

into the equation

ax + by = c

yields

(-10)(6) + (15)(5) = 15

, so the value of

c

is

15

.

Step-by-Step Solution

1

Rewrite the standard form equation

ax + by = c

in slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

This allows the identification of the slope of the line in terms of the coefficients

a

and

b

.

2

Set the slope expression equal to the given slope of

\frac{2}{3}

and solve for

b

in terms of

a

.

b = -1.5a

The slope of the line is

-\frac{a}{b}

, so

-\frac{a}{b} = \frac{2}{3} \implies 2b = -3a \implies b = -1.5a

.

3

Substitute

b = -1.5a

into the given equation

a + b = 5

and solve for

a

, then find

b

.

a = -10

and

b = 15

Substituting gives

a - 1.5a = 5 \implies -0.5a = 5 \implies a = -10

. Substituting

a = -10

back into the relationship gives

b = -1.5(-10) = 15

.

4

Substitute

a = -10

,

b = 15

, and the point

(6, 5)

into the equation

ax + by = c

and solve for

c

.

c = 15

(-10)(6) + (15)(5) = c \implies -60 + 75 = 15 \implies c = 15

.

Key Concept

Linear Equations in Two Variables

Question 28Question →

The table below shows several values of $x$ and their corresponding values of $y$ for a linear relationship, where $p$ and $k$ are constants.

$x$	$y$
$p$	$4$
$p + 2$	$10$
$2p + 3$	$k$

If $k = 25$ , what is the value of $p$ ?

0.5

4

6

60

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. A linear relationship has a constant rate of change (slope). Calculating the slope

m

from the first two points

(p, 4)

and

(p + 2, 10)

gives

m = \frac{10 - 4}{(p + 2) - p} = 3

. The equation of the line is

y - 4 = 3(x - p)

, which simplifies to

y = 3x - 3p + 4

. Substituting the third point

(2p + 3, 25)

into this equation gives

25 = 3(2p + 3) - 3p + 4

, which simplifies to

25 = 3p + 13

. Solving for

p

yields

p = 4

.

Step-by-Step Solution

1

Determine the slope of the linear relationship using the first two points from the table,

(p, 4)

and

(p + 2, 10)

.

The slope is

m = 3

.

A linear relationship has a constant rate of change (slope), which is the change in

y

divided by the change in

x

:

m = \frac{10 - 4}{(p + 2) - p} = \frac{6}{2} = 3

.

2

Write the equation of the line using the point-slope form with the point

(p, 4)

and slope

m = 3

.

y = 3x - 3p + 4

Expressing the linear relationship as an equation allows us to find the relationship between the parameters

p

and

k

:

y - 4 = 3(x - p) \Rightarrow y = 3x - 3p + 4

.

3

Substitute the third point

(2p + 3, k)

into the linear equation and simplify the expression to solve for

k

in terms of

p

.

k = 3p + 13

Since the point

(2p + 3, k)

lies on the line, its coordinates must satisfy the line's equation:

k = 3(2p + 3) - 3p + 4 \Rightarrow k = 6p + 9 - 3p + 4 \Rightarrow k = 3p + 13

.

4

Substitute the given value

k = 25

into the equation

k = 3p + 13

and solve for

p

.

p = 4

This isolates the variable

p

to find its numerical value under the specified conditions:

25 = 3p + 13 \Rightarrow 12 = 3p \Rightarrow p = 4

.

Key Concept

Finding the equation of a line from a table of values and using substitution to solve for unknown parameters in a linear relationship.

Alternative Method

An alternative method is to use the property that the slope between any two points on a line is constant. We can equate the slope between

(p, 4)

and

(p + 2, 10)

to the slope between

(p, 4)

and

(2p + 3, 25)

. This gives the equation

\frac{10 - 4}{(p + 2) - p} = \frac{25 - 4}{(2p + 3) - p}

, which simplifies to

3 = \frac{21}{p + 3}

. Solving this equation yields

3(p + 3) = 21

, so

p + 3 = 7

, which gives

p = 4

.

Estimated Time:2m 0s

Question 29Question →

A line has the equation $y = mx + k$ in the $xy$ -plane, where $m$ and $k$ are constants. If the line passes through the points $(2, p)$ and $(p, -4)$ , and has a $y$ -intercept of $(0, 3p)$ for some constant $p > 0$ , what is the slope, $m$ , of the line?

-4

4

-1

1

Show answer & explanation

Answer: $-4$

Answer

-4

The correct answer is

-4

. Substituting the

y

-intercept

(0, 3p)

into the slope-intercept form equation

y = mx + k

gives

k = 3p

. Substituting the point

(2, p)

into

y = mx + 3p

gives

p = 2m + 3p

, which simplifies to

m = -p

. Then, substituting the point

(p, -4)

into

y = mx + 3p

gives

-4 = mp + 3p

. Replacing

m

with

-p

results in

-4 = -p^2 + 3p

, which can be rearranged to the quadratic equation

p^2 - 3p - 4 = 0

. Factoring this equation yields

(p - 4)(p + 1) = 0

. Since

p > 0

, the only valid solution is

p = 4

. Substituting

p = 4

back into the expression for

m

gives the slope

m = -4

.

Step-by-Step Solution

1

Determine the y-intercept constant

k

in terms of

p

.

k = 3p

The equation of the line is

y = mx + k

. Since the

y

-intercept is given as

(0, 3p)

, substituting

x = 0

and

y = 3p

gives

k = 3p

.

2

Find the relationship between the slope

m

and the parameter

p

.

m = -p

Since the line passes through

(2, p)

, we substitute these coordinates into the equation

y = mx + 3p

to get

p = 2m + 3p

. Solving for

m

yields

2m = -2p

, or

m = -p

.

3

Substitute the point

(p, -4)

into the linear equation and solve the resulting quadratic equation for

p

.

p = 4

Substituting the coordinates of

(p, -4)

into

y = mx + 3p

gives

-4 = mp + 3p

. Substituting

m = -p

gives

-4 = -p^2 + 3p

, which simplifies to the quadratic equation

p^2 - 3p - 4 = 0

. Factoring this equation gives

(p - 4)(p + 1) = 0

. Since

p > 0

, we must have

p = 4

.

4

Calculate the slope

m

.

m = -4

Since

m = -p

and we found

p = 4

, the slope is

m = -4

.

Key Concept

Determining the slope of a line from given points and a parameter in a linear equation.

Alternative Method

Instead of finding the equation of the line first, the slope

m

can be expressed directly using the slope formula between the points

(2, p)

and

(p, -4)

:

m = \frac{-4 - p}{p - 2}

. Since the slope of the line connecting

(0, 3p)

and

(2, p)

is also

m = \frac{p - 3p}{2 - 0} = -p

, we can set these two expressions for

m

equal to each other:

-p = \frac{-4 - p}{p - 2}

. Multiplying both sides by

p - 2

yields

-p(p - 2) = -4 - p \implies -p^2 + 2p = -4 - p

. Rearranging the terms gives the quadratic equation

p^2 - 3p - 4 = 0

. Factoring gives

(p - 4)(p + 1) = 0

. Since

p > 0

, we find

p = 4

, and the slope is

m = -p = -4

.

Estimated Time:2m 0s

Question 30Question →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, has the property that the value of $y$ decreases by $3$ for every increase of $5$ in the value of $x$ . If the graph of this equation passes through the point $(4, -2)$ and $a = 6$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is

4

.

The correct answer is

4

. The slope of the line described is

m = \frac{\Delta y}{\Delta x} = \frac{-3}{5} = -0.6

. Rearranging

ax + by = c

into slope-intercept form

y = -\frac{a}{b}x + \frac{c}{b}

shows that the slope is

-\frac{a}{b}

. Setting these equal gives

-\frac{a}{b} = -\frac{3}{5} \implies \frac{a}{b} = \frac{3}{5}

. Since

a = 6

, we find

b = 10

. Substituting

a = 6

,

b = 10

, and the coordinates of the point

(4, -2)

into the original equation yields

6(4) + 10(-2) = 4

.

Step-by-Step Solution

1

Determine the slope of the line from the described relationship between

x

and

y

.

The slope

m

of the line is

-\frac{3}{5}

.

The slope represents the change in

y

divided by the change in

x

. Since

y

decreases by

3

(

\Delta y = -3

) for every increase of

5

in

x

(

\Delta x = 5

), the slope is

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

.

2

Express the slope of the line in terms of the coefficients from the standard form equation

ax + by = c

.

The slope of the line is

-\frac{a}{b}

.

Rewriting the equation

ax + by = c

in slope-intercept form gives

by = -ax + c \implies y = -\frac{a}{b}x + \frac{c}{b}

. The coefficient of

x

is the slope, so

m = -\frac{a}{b}

.

3

Equate the two expressions for the slope and solve for

b

using the given value of

a = 6

.

b = 10

Setting

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

gives

\frac{a}{b} = \frac{3}{5}

. Substituting

a = 6

yields

\frac{6}{b} = \frac{3}{5}

, which simplifies to

3b = 30

, so

b = 10

.

4

Substitute the point

(4, -2)

and the values of

a

and

b

into the equation

ax + by = c

to solve for

c

.

c = 4

Substituting

a = 6

,

b = 10

,

x = 4

, and

y = -2

into

ax + by = c

gives

6(4) + 10(-2) = c \implies 24 - 20 = c \implies c = 4

.

Key Concept

Linear Equations in Two Variables

Question 31Question →

A water utility company charges a flat monthly connection fee plus a constant rate per gallon of water consumed. In a certain month, a household that consumed $2,800$ gallons of water was billed a total of $62.00$ dollars, and a household that consumed $4,500$ gallons of water was billed a total of $87.50$ dollars. What is the flat monthly connection fee, in dollars, charged by the water utility company?

Show answer & explanation

Answer: 20

Answer

The flat monthly connection fee charged by the water utility company is 20 dollars.

The flat monthly connection fee is 20 dollars. A linear relationship can be modeled by the equation

y = mx + b

, where

x

represents the water consumed in gallons,

y

represents the total bill in dollars,

m

represents the rate per gallon, and

b

represents the flat monthly connection fee. Using the data for the two households, we get the points

(2800, 62)

and

(4500, 87.5)

. The rate per gallon is the slope of the line passing through these points:

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon. Substituting this slope and the point

(2800, 62)

into the equation gives

62 = 0.015(2800) + b

, which simplifies to

62 = 42 + b

. Solving for

b

yields

20

dollars.

Step-by-Step Solution

1

Define the linear relationship using the variables

x

for water consumed in gallons and

y

for the total bill in dollars.

The relationship can be written as

y = mx + b

, where

m

represents the cost per gallon of water and

b

represents the flat monthly connection fee.

This establishes the linear framework needed to solve for the unknown constants using the provided data points.

2

Calculate the constant rate of change (slope

m

) using the coordinates of the two households:

(2800, 62)

and

(4500, 87.5)

.

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon.

The slope of a linear function represents the constant rate of change between the two variables.

3

Substitute the calculated slope

m = 0.015

and the point

(2800, 62)

into the equation

y = mx + b

to find the flat fee

b

.

62 = 0.015(2800) + b \implies 62 = 42 + b \implies b = 20

.

The flat monthly connection fee corresponds to the vertical intercept (

y

-intercept) of the linear equation.

Key Concept

Linear Equations in Two Variables

Question 32Question →

In the $xy$ -plane, a line has a $y$ -intercept of $(0, d)$ and passes through the point $(4, d - 3)$ , where $d$ is a constant. If the line also passes through the point $(12, k)$ , which of the following expressions represents the value of $k$ ?

d + 9

d - 16

d - 9

d - 36

Show answer & explanation

Answer: $d - 9$

Answer

d - 9

The correct answer represents the value of

k

as

d - 9

. The slope

m

of the line passing through

(0, d)

and

(4, d - 3)

is

\frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

. Using the

y

-intercept

(0, d)

, the equation of the line is

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

. Substituting the point

(12, k)

gives

k = -\frac{3}{4}(12) + d = d - 9

.

Step-by-Step Solution

1

Calculate the slope of the line using the points

(0, d)

and

(4, d - 3)

.

m = -\frac{3}{4}

The slope formula is

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

. Substituting the given coordinates yields

m = \frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

.

2

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

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

Since the line's

y

-intercept is

(0, d)

, the constant

b

in the slope-intercept equation

y = mx + b

is equal to

d

.

3

Substitute the point

(12, k)

into the equation of the line to solve for

k

.

k = d - 9

Substituting

x = 12

and

y = k

into

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

gives

k = -\frac{3}{4}(12) + d = -9 + d

, which simplifies to

d - 9

.

Key Concept

Determining the equation and coordinates of a line in the coordinate plane given its slope and intercepts.

Question 33Question →

In the $xy$ -plane, the graph of the linear equation $3x + 5y = c$ , where $c$ is a constant, passes through the point $(4, 3)$ . What is the $x$ -coordinate of the point on this graph where the $y$ -coordinate is $-3$ ?

Show answer & explanation

Answer: 14

Answer

The correct answer is 14.

To find the correct answer, first substitute the given point

(4, 3)

into the equation

3x + 5y = c

to determine the value of the constant

c

:

3(4) + 5(3) = 12 + 15 = 27

. Thus, the equation is

3x + 5y = 27

. Next, substitute

-3

for

y

in this equation to find the corresponding

x

-coordinate:

3x + 5(-3) = 27 \implies 3x - 15 = 27

. Adding

15

to both sides gives

3x = 42

, and dividing by

3

yields

x = 14

.

Step-by-Step Solution

1

Substitute the point

(4, 3)

into the equation

3x + 5y = c

to solve for

c

.

c = 27

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

2

Substitute

y = -3

and

c = 27

into the equation

3x + 5y = c

.

3x - 15 = 27

We want to find the

x

-coordinate of the point on the line when the

y

-coordinate is

-3

.

3

Solve the equation

3x - 15 = 27

for

x

.

x = 14

Isolating

x

gives the

x

-coordinate of the point.

Key Concept

Using a known point on a line to find a constant coefficient or constant term, and using the resulting equation to find other coordinates.

Alternative Method

Alternatively, you can write the equation in slope-intercept form. Solving

3x + 5y = c

for

y

gives

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

. The slope of the line is

-\frac{3}{5}

. Since the slope is constant, the change in

y

divided by the change in

x

between

(4, 3)

and

(x, -3)

is equal to the slope:

\frac{-3 - 3}{x - 4} = -\frac{3}{5} \implies \frac{-6}{x - 4} = -\frac{3}{5}

. Cross-multiplying gives

-30 = -3(x - 4) \implies 10 = x - 4 \implies x = 14

.

Estimated Time:1m 30s

Question 34Question →

In the $xy$ -plane, the graph of the linear equation $y - 4 = m(x - 6)$ , where $m$ is a constant, has a $y$ -intercept of $(0, 7)$ . What is the $x$ -coordinate of the $x$ -intercept of the line?

-4

-2

14

8

Show answer & explanation

Answer: 14

Answer

The correct answer is 14. This is the x-coordinate of the point where the line crosses the x-axis, which is (14, 0).

To find the value of

m

, substitute the y-intercept

(0, 7)

into the equation:

7 - 4 = m(0 - 6)

, which simplifies to

3 = -6m

, so

m = -\frac{1}{2}

. The equation of the line is

y - 4 = -\frac{1}{2}(x - 6)

. The x-intercept occurs when

y = 0

. Substituting

y = 0

gives

-4 = -\frac{1}{2}(x - 6)

, which simplifies to

8 = x - 6

, so

x = 14

.

Step-by-Step Solution

1

Substitute the y-intercept

(0, 7)

into the given equation to solve for the constant

m

.

7 - 4 = m(0 - 6) \implies 3 = -6m \implies m = -\frac{1}{2}

The y-intercept point

(0, 7)

must satisfy the equation of the line, allowing us to find the slope

m

.

2

Write the linear equation using the calculated value of

m

.

y - 4 = -\frac{1}{2}(x - 6)

Knowing the slope allows us to write the complete linear equation in point-slope form.

3

Find the x-intercept of the line by setting

y = 0

and solving for

x

.

0 - 4 = -\frac{1}{2}(x - 6) \implies -4 = -\frac{1}{2}(x - 6) \implies 8 = x - 6 \implies x = 14

The x-intercept is the point on the line where the y-coordinate is equal to 0.

Key Concept

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

Alternative Method

Alternatively, write the equation in slope-intercept form

y = mx + b

. Since the y-intercept is

(0, 7)

, we have

b = 7

, so the equation is

y = mx + 7

. Since the line passes through the point

(6, 4)

(determined by the point-slope form

y - 4 = m(x - 6)

), substitute this point into the equation:

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

. Thus, the equation is

y = -\frac{1}{2}x + 7

. Setting

y = 0

for the x-intercept gives

0 = -\frac{1}{2}x + 7 \implies \frac{1}{2}x = 7 \implies x = 14

.

Estimated Time:1m 30s

Question 35Question →

A manufacturer plans to produce two models of smartphones: Model X and Model Y. The assembly of each Model X smartphone requires $2.5$ hours, and the assembly of each Model Y smartphone requires $4$ hours. A total of $800$ hours of assembly time is allocated for producing these two models. If the manufacturer produces $120$ Model X smartphones, what is the maximum number of Model Y smartphones that can be produced using the remaining allocated assembly time?

125

128

150

275

Show answer & explanation

Answer: 125

Answer

The maximum number of Model Y smartphones that can be produced is 125.

The correct answer is the value that satisfies the linear relation under the given constraints. Setting up the equation

2.5x + 4y = 800

and substituting

120

for

x

results in

300 + 4y = 800

. Subtracting

300

from both sides yields

4y = 500

, and dividing by

4

gives

125

Model Y smartphones.

Step-by-Step Solution

1

Set up a linear equation representing the total assembly time.

2.5x + 4y = 800

, where

x

is the number of Model X smartphones and

y

is the number of Model Y smartphones.

The total assembly time is the sum of the time spent on Model X (

2.5

hours per phone) and Model Y (

4

hours per phone), which must equal the allocated

800

hours.

2

Substitute the given number of Model X smartphones into the equation.

2.5(120) + 4y = 800

, which simplifies to

300 + 4y = 800

.

The problem states that the manufacturer produces

120

Model X smartphones.

3

Isolate the variable representing Model Y smartphones to find its value.

4y = 500

, which simplifies to

y = 125

.

Subtracting

300

from both sides of the equation and then dividing by

4

isolates the variable

y

.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 36Question →

During a chemistry experiment, the temperature of a liquid increases at a constant rate. The temperature of the liquid is $24^\circ\text{C}$ at $5$ minutes after the experiment starts, and $48^\circ\text{C}$ at $15$ minutes after the experiment starts. The temperature $T$ , in degrees Celsius, of the liquid $t$ minutes after the experiment starts can be modeled by the equation $T = mt + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. By translating the given information into two points on the line,

(5, 24)

and

(15, 48)

, we can find the slope

m = \frac{48 - 24}{15 - 5} = 2.4

. Substituting this slope and the point

(5, 24)

into

T = mt + b

yields

24 = 2.4(5) + b

, which simplifies to

24 = 12 + b

. Solving for

b

gives

b = 12

.

Step-by-Step Solution

1

Identify the data points representing the relationship between time and temperature.

The two data points are

(t_1, T_1) = (5, 24)

and

(t_2, T_2) = (15, 48)

.

These points represent coordinates

(t, T)

on the line representing the temperature over time.

2

Calculate the slope

m

of the linear equation.

m = \frac{48 - 24}{15 - 5} = \frac{24}{10} = 2.4

The slope represents the constant rate of temperature increase per minute.

3

Substitute the slope

m

and the coordinates of one point into the equation

T = mt + b

to solve for the y-intercept

b

.

24 = 2.4(5) + b \implies 24 = 12 + b \implies b = 12

This determines the value of the constant

b

, which corresponds to the initial temperature of the liquid.

Key Concept

Determining the y-intercept of a linear model in two variables given two points.

Question 37Question →

The graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line in the $xy$ -plane. If this line passes through the points $(2, 9)$ and $(6, 3)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is

5

.

Substituting the coordinates of the two points

(2, 9)

and

(6, 3)

into the given linear equation

ax + by = 24

yields a system of two equations:

2a + 9b = 24

and

6a + 3b = 24

. Simplifying the second equation gives

2a + b = 8

, which means

b = 8 - 2a

. Substituting this expression for

b

into the first equation yields

2a + 9(8 - 2a) = 24

, which simplifies to

-16a = -48

, or

a = 3

. Plugging

a = 3

back into

b = 8 - 2a

gives

b = 2

. Thus, the value of

a + b

is

3 + 2 = 5

.

Step-by-Step Solution

1

Substitute the point

(2, 9)

into the equation

ax + by = 24

.

2a + 9b = 24

Since the line passes through the point

(2, 9)

, the coordinates must satisfy the equation of the line.

2

Substitute the point

(6, 3)

into the equation

ax + by = 24

.

6a + 3b = 24

Since the line passes through the point

(6, 3)

, the coordinates must satisfy the equation of the line.

3

Solve the system of equations for

a

and

b

.

a = 3

and

b = 2

To find the values of the constants

a

and

b

, we solve the linear system: (1)

2a + 9b = 24

and (2)

6a + 3b = 24

. Dividing the second equation by 3 gives

2a + b = 8

, or

b = 8 - 2a

. Substituting this into the first equation gives

2a + 9(8 - 2a) = 24 \implies 2a + 72 - 18a = 24 \implies -16a = -48 \implies a = 3

. Then,

b = 8 - 2(3) = 2

.

4

Add the values of

a

and

b

.

a + b = 5

The question asks for the value of

a + b

.

Key Concept

Solving systems of linear equations derived from coordinate substitution in a two-variable linear equation.

Estimated Time:1m 30s

Question 38Question →

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

Show answer & explanation

Answer: 10

Answer

10

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

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

kWh per hour. The relationship between energy

E

and hours

h

can be written as

E = 3h + b

. Substituting the known point

(3, 14)

gives

14 = 3(3) + b

, which simplifies to

b = 5

. Thus, the linear equation representing this relationship is

E = 3h + 5

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

35

kWh, set

E = 35

in the equation:

35 = 3h + 5

. Subtracting 5 from both sides yields

30 = 3h

, and dividing by 3 gives

h = 10

hours.

Step-by-Step Solution

1

Calculate the rate of energy storage per hour.

3 kWh per hour

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

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

.

2

Determine the initial energy stored at sunrise.

5 kWh

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

14 - 3(3) = 5

.

3

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

10 hours

The linear relationship is

E = 3h + 5

. Setting

E = 35

gives

35 = 3h + 5

. Subtracting 5 from both sides gives

30 = 3h

, which simplifies to

h = 10

.

Key Concept

Linear Equations in Two Variables

Question 39Question →

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

Show answer & explanation

Answer: 4

Answer

The value of

b

is

4

.

The linear equation

3x + by = 36

can be rewritten in slope-intercept form by subtracting

3x

from both sides to get

by = -3x + 36

, and then dividing all terms by

b

to get

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

. The slope of this line is the coefficient of

x

, which is

-\frac{3}{b}

. Setting this equal to the given slope of

-0.75

(or

-\frac{3}{4}

) gives

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

. Solving for

b

yields

b = 4

.

Step-by-Step Solution

1

Express the given linear equation in slope-intercept form.

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

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

b

, we rewrite the equation

3x + by = 36

in the form

y = mx + d

.

2

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

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

The coefficient of

x

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

-0.75

.

3

Solve the equation for

b

.

b = 4

Multiply both sides of the equation by

-1

to get

\frac{3}{b} = 0.75

. Since

0.75 = \frac{3}{4}

, we have

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

, which gives

b = 4

.

Key Concept

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

Estimated Time:1m 30s

Question 40Question →

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

-10

-6

-4

5

Show answer & explanation

Answer: -10

Answer

-10

The correct answer is

-10

. By substituting the point

(6, 8)

into the equation

y - 4 = m(x + 2)

, we obtain

8 - 4 = m(6 + 2)

, which simplifies to

4 = 8m

, giving

m = 0.5

. Using this value of

m

, the equation of the line is

y - 4 = 0.5(x + 2)

. The

x

-intercept is the value of

x

when

y = 0

. Substituting

0

for

y

gives

-4 = 0.5(x + 2)

, which simplifies to

-8 = x + 2

. Solving for

x

gives

x = -10

.

Step-by-Step Solution

1

Substitute the given point

(6, 8)

into the equation to find the value of the constant

m

.

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

Since the line passes through the point

(6, 8)

, these coordinates must satisfy the equation.

2

Write the completed equation of the line using the value of

m

.

y - 4 = 0.5(x + 2)

Substituting

m = 0.5

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

3

Find the

x

-intercept by setting

y = 0

and solving for

x

.

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

The

x

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

x

-axis, which occurs where

y = 0

.

Key Concept

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

Linear Equations in Two Variables

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution