All practice questions

In the system of equations below, what is the value of $x$ ?

3x - y = 7

x + y = 5

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

Adding the two equations eliminates

y

, resulting in

4x = 12

. Dividing by

4

gives

x = 3

.

Step-by-Step Solution

1

Add the two equations to eliminate

y

.

4x = 12

Adding the equations

(3x - y) + (x + y) = 7 + 5

eliminates

y

because

-y + y = 0

.

2

Solve for

x

.

x = 3

Divide both sides of the equation

4x = 12

by

4

.

Key Concept

Solving a system of linear equations by elimination.

Question 62Question →

If $\frac{3}{5}p - 4 = 8$ , what is the value of $p$ ?

Show answer & explanation

Answer: 20

Answer

20

To solve the equation

\frac{3}{5}p - 4 = 8

, we perform inverse operations to isolate

p

. First, add 4 to both sides of the equation to get

\frac{3}{5}p = 12

. Next, multiply both sides by the reciprocal of the coefficient of

p

, which is

\frac{5}{3}

. This gives

p = 12 \times \frac{5}{3} = 20

. Substituting 20 back into the original equation confirms it is the correct solution.

Step-by-Step Solution

1

Add 4 to both sides of the equation.

\frac{3}{5}p = 12

To isolate the term containing the variable

p

.

2

Multiply both sides of the equation by

\frac{5}{3}

.

p = 20

To solve for

p

by multiplying by the reciprocal of its coefficient.

Key Concept

Solving one-variable linear equations using inverse operations.

Question 63Question →

In the equation below, $k$ is a constant.

\frac{1}{3}(2kx - 9) - \frac{5}{6}(x + 4) = \frac{11}{2}

If the equation has no solution, what is the value of $k$ ?

Show answer & explanation

Answer: 1.25

Answer

1.25 (or 5/4)

The correct answer is 1.25 (or 5/4). A linear equation in one variable of the form

Ax + B = C

has no solution if the variable terms on both sides of the equation are equal (meaning

A = 0

) and the constant terms are unequal (

B \neq C

). Expanding the left side of the given equation yields

\frac{2}{3}kx - 3 - \frac{5}{6}x - \frac{10}{3} = \frac{11}{2}

. Combining the constant terms gives

\left(\frac{2}{3}k - \frac{5}{6}\right)x - \frac{19}{3} = \frac{11}{2}

. Setting the coefficient of

x

to

0

yields

\frac{2}{3}k - \frac{5}{6} = 0

. Solving for

k

gives

k = \frac{5}{6} \times \frac{3}{2} = \frac{5}{4}

, which is equivalent to 1.25. Since the remaining constant terms are unequal (

-\frac{19}{3} \neq \frac{11}{2}

), the equation has no solution when

k = 1.25

.

Step-by-Step Solution

1

Expand the expression on the left side of the equation

\frac{2}{3}kx - 3 - \frac{5}{6}x - \frac{10}{3} = \frac{11}{2}

Apply the distributive property to remove the parentheses.

2

Group the

x

terms and combine the constants on the left side

\left(\frac{2}{3}k - \frac{5}{6}\right)x - \frac{19}{3} = \frac{11}{2}

Simplify the equation by combining like terms:

-3 - \frac{10}{3} = -\frac{9}{3} - \frac{10}{3} = -\frac{19}{3}

.

3

Set the coefficient of the

x

term equal to

0

\frac{2}{3}k - \frac{5}{6} = 0

For a linear equation to have no solution, the variable terms on both sides of the equation must cancel out (meaning the coefficient of the variable must be

0

), while the remaining constant terms must not be equal (

-\frac{19}{3} \neq \frac{11}{2}

).

4

Solve the resulting equation for

k

k = 1.25

Add

\frac{5}{6}

to both sides to get

\frac{2}{3}k = \frac{5}{6}

, then multiply by the reciprocal of

\frac{2}{3}

, which gives

k = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1.25

.

Key Concept

Conditions for a linear equation in one variable to have no solution

Question 64Question →

A rectangle has a length of $2x + 5$ centimeters and a width of $x - 2$ centimeters. If the perimeter of the rectangle is $42$ centimeters, what is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. The perimeter of a rectangle is calculated using the formula

P = 2l + 2w

. Substituting the given expressions for length and width yields the linear equation

2(2x + 5) + 2(x - 2) = 42

. Distributing the 2 gives

4x + 10 + 2x - 4 = 42

. Combining like terms on the left side simplifies this to

6x + 6 = 42

. Subtracting 6 from both sides yields

6x = 36

. Finally, dividing both sides by 6 gives the value

x = 6

.

Step-by-Step Solution

1

Write the perimeter equation in terms of

x

.

2(2x + 5) + 2(x - 2) = 42

The perimeter of a rectangle is the sum of all its sides, represented by the formula

P = 2l + 2w

.

2

Distribute the 2 into the parentheses.

4x + 10 + 2x - 4 = 42

Applying the distributive property yields

2(2x + 5) = 4x + 10

and

2(x - 2) = 2x - 4

.

3

Combine like terms on the left side.

6x + 6 = 42

Grouping the variable terms gives

4x + 2x = 6x

, and grouping the constant terms gives

10 - 4 = 6

.

4

Subtract 6 from both sides of the equation.

6x = 36

This isolates the variable term on the left side of the equation.

5

Divide both sides of the equation by 6.

x = 6

Dividing by the coefficient of

x

solves for the variable.

Key Concept

Solving linear equations in one variable by applying the distributive property and combining like terms.

Question 65Question →

In the equation below, $c$ is a constant.

\frac{3c(2x - 1) - 2(x + 4c)}{5} = 2x - 3

If the equation has no solution for $x$ , what is the value of $c$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

To find the value of

c

that results in no solution, we first clear the fraction by multiplying both sides of the equation by 5, giving

3c(2x - 1) - 2(x + 4c) = 10x - 15

. Next, we expand the terms to get

6cx - 3c - 2x - 8c = 10x - 15

, and group the

x

terms and constant terms:

(6c - 2)x - 11c = 10x - 15

. For a linear equation to have no solution, the coefficients of

x

on both sides must be equal while the constant terms must be different. Setting the coefficients equal gives

6c - 2 = 10

, which solves to

c = 2

. Checking the constant terms when

c = 2

, we get

-11(2) = -22

on the left and

-15

on the right. Since

-22 \neq -15

, the equation has no solution, confirming

c = 2

is correct.

Step-by-Step Solution

1

Multiply both sides of the equation by 5 to eliminate the denominator.

3c(2x - 1) - 2(x + 4c) = 10x - 15

Clearing the denominator simplifies the equation into standard polynomial terms.

2

Distribute the terms on the left side of the equation.

6cx - 3c - 2x - 8c = 10x - 15

Applying the distributive property expands the expression so terms can be grouped.

3

Group the

x

terms and constant terms on the left side.

(6c - 2)x - 11c = 10x - 15

Putting the equation in the standard form

Ax + B = Cx + D

allows us to easily set up the conditions for no solution.

4

Set the coefficients of

x

on both sides equal to each other.

6c - 2 = 10

, which simplifies to

6c = 12

, and thus

c = 2

.

For the equation to have no solution, the variable terms on both sides must cancel each other out.

5

Verify that the constant terms are not equal when

c = 2

.

The left-side constant is

-11(2) = -22

, and the right-side constant is

-15

. Since

-22 \neq -15

, the equation has no solution.

If the constant terms were equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Identifying conditions for a linear equation in one variable to have no solution.

Alternative Method

Instead of clearing the fraction first, write the left side of the equation as

(\frac{6c - 2}{5})x - \frac{11c}{5}

. For there to be no solution, the coefficient of

x

on the left side,

\frac{6c - 2}{5}

, must equal the coefficient of

x

on the right side, which is 2. Solving

\frac{6c - 2}{5} = 2

gives

6c - 2 = 10 \implies c = 2

.

Estimated Time:2m 30s

Question 66Question →

A linear function $f$ has a slope of $\frac{2}{3}$ . The function $g$ is defined by $g(x) = f(x + 4) - 3$ . If the graph of $y = g(x)$ in the $xy$ -plane has an $x$ -intercept at $(5, 0)$ , what is the $x$ -intercept of the graph of $y = f(x)$ ?

Show answer & explanation

Answer: 4.5

Answer

4.5

The correct answer is

4.5

(or

\frac{9}{2}

). By using the function transformation relation, we find

g(5) = f(9) - 3 = 0

, which yields

f(9) = 3

. With a slope of

\frac{2}{3}

, the equation of the line is

f(x) = \frac{2}{3}x - 3

. Setting

f(x) = 0

yields the

x

-intercept at

x = 4.5

.

Step-by-Step Solution

1

Use the definition of function

g

and its given

x

-intercept to find a point on the graph of

f

.

f(9) = 3

Since the graph of

y = g(x)

has an

x

-intercept at

(5, 0)

, we know

g(5) = 0

. Substituting

x = 5

into the definition

g(x) = f(x + 4) - 3

gives

g(5) = f(5 + 4) - 3 = f(9) - 3

. Since

g(5) = 0

, it follows that

f(9) - 3 = 0

, or

f(9) = 3

.

2

Determine the equation of the linear function

f

using its slope and the point identified in Step 1.

f(x) = \frac{2}{3}x - 3

The function

f

is linear with a slope of

\frac{2}{3}

and passes through the point

(9, 3)

. Using the point-slope formula, we get

f(x) - 3 = \frac{2}{3}(x - 9)

, which simplifies to

f(x) = \frac{2}{3}x - 3

.

3

Find the

x

-intercept of the graph of

y = f(x)

by setting

f(x) = 0

.

x = 4.5

(or

\frac{9}{2}

)

To find the

x

-intercept, set

f(x) = 0

. This gives

\frac{2}{3}x - 3 = 0

. Adding

3

to both sides and multiplying by

\frac{3}{2}

yields

x = \frac{9}{2}

, which is equal to

4.5

.

Key Concept

Linear Functions and Graphs

Alternative Method

Alternatively, we can write the equation of

f

in slope-intercept form as

f(x) = \frac{2}{3}x + b

. Then the definition of

g(x)

becomes

g(x) = \frac{2}{3}(x + 4) + b - 3 = \frac{2}{3}x + \frac{8}{3} + b - 3 = \frac{2}{3}x + b - \frac{1}{3}

. Since the graph of

g

has an

x

-intercept at

(5, 0)

, we substitute

x = 5

and

g(5) = 0

to get

\frac{2}{3}(5) + b - \frac{1}{3} = 0

, which simplifies to

3 + b = 0

, so

b = -3

. This gives

f(x) = \frac{2}{3}x - 3

. Finally, the

x

-intercept is found by solving

\frac{2}{3}x - 3 = 0

, resulting in

x = 4.5

.

Estimated Time:2m 0s

Question 67Question →

If $\frac{2}{3}(x - 4) = 6$ , what is the value of $x$ ?

Show answer & explanation

Answer: 13

Answer

13

To solve the linear equation

\frac{2}{3}(x - 4) = 6

, multiply both sides of the equation by

\frac{3}{2}

to isolate

x - 4

. This yields

x - 4 = 9

. Adding

4

to both sides of the equation isolates

x

and gives the value

x = 13

.

Step-by-Step Solution

1

Multiply both sides of the equation

\frac{2}{3}(x - 4) = 6

by the reciprocal of the fraction, which is

\frac{3}{2}

.

x - 4 = 9

Multiplying a fraction by its reciprocal simplifies it to 1, leaving the term in parentheses isolated on the left side.

2

Add

4

to both sides of the equation

x - 4 = 9

.

x = 13

Adding

4

isolates the variable

x

on the left side.

Key Concept

Solving a one-variable linear equation using inverse operations.

Question 68Question →

A food truck sells tacos and burritos. On Tuesday, the food truck sold a total of 120 tacos and burritos. The number of tacos sold, $t$ , was 20 more than the number of burritos sold, $b$ . This situation is represented by the system of equations below:

t + b = 120

t = b + 20

How many burritos did the food truck sell on Tuesday?

Show answer & explanation

Answer: 50

Answer

The food truck sold 50 burritos on Tuesday.

To find the number of burritos sold,

b

, we substitute

t = b + 20

into the first equation

t + b = 120

, giving

(b + 20) + b = 120

. Combining like terms yields

2b + 20 = 120

. Subtracting 20 from both sides results in

2b = 100

. Finally, dividing both sides by 2 gives

b = 50

. Thus, the food truck sold 50 burritos on Tuesday.

Step-by-Step Solution

1

Substitute the expression for

t

from the second equation into the first equation

(b + 20) + b = 120

To eliminate the variable

t

and set up an equation with only one variable,

b

.

2

Combine like terms on the left side of the equation

2b + 20 = 120

To simplify the equation for solving.

3

Subtract 20 from both sides of the equation

2b = 100

To isolate the variable term on one side of the equation.

4

Divide both sides of the equation by 2

b = 50

To solve for the variable

b

representing the number of burritos.

Key Concept

Solving systems of linear equations using the substitution method

Question 69Question →

In the equation $3(4x + b) - 2(x - 5) = 2(5x + 8)$ , $b$ is a constant. If the equation has infinitely many solutions for $x$ , what is the value of $b$ ?

Show answer & explanation

Answer: 2

Answer

The value of

b

is 2.

Distributing the constants in the equation

3(4x + b) - 2(x - 5) = 2(5x + 8)

yields

12x + 3b - 2x + 10 = 10x + 16

. Combining like terms on the left side simplifies the equation to

10x + 3b + 10 = 10x + 16

. For a linear equation in one variable to have infinitely many solutions, both sides of the equation must be identical. Since the coefficients of

x

are equal (

10 = 10

), the constant terms must also be equal:

3b + 10 = 16

. Solving for

b

yields

3b = 6

, which simplifies to

b = 2

.

Step-by-Step Solution

1

Distribute the constants through the parentheses on both sides of the equation

3(4x + b) - 2(x - 5) = 2(5x + 8)

.

12x + 3b - 2x + 10 = 10x + 16

To eliminate parentheses and allow grouping of like terms.

2

Combine like terms on the left side of the equation.

10x + 3b + 10 = 10x + 16

To simplify the left-hand expression into the standard linear form.

3

Equate the constant terms on both sides of the equation.

3b + 10 = 16

A linear equation in one variable has infinitely many solutions when both sides are identical. Since the coefficients of the variable

x

are both 10, the constant terms must be equal.

4

Solve for

b

by isolating it.

b = 2

Subtracting 10 from both sides gives

3b = 6

, and dividing by 3 yields the final value.

Key Concept

Determining conditions for a linear equation in one variable to have infinitely many solutions (identity).

Question 70Question →

In the $xy$ -plane, line $l$ passes through the points $(-2, -3)$ and $(2, 5)$ . Line $k$ is parallel to line $l$ and has a $y$ -intercept of $(0, -1)$ . If the point $(a, 9)$ lies on line $k$ , what is the value of $a$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a

is

5

.

To find the value of

a

, we first determine the slope of line

l

using the two given points,

(-2, -3)

and

(2, 5)

. The slope

m

is given by

\frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2

. Since line

k

is parallel to line

l

, it has the same slope of

2

. The

y

-intercept of line

k

is

(0, -1)

, so the equation of line

k

is

y = 2x - 1

. To find the value of

a

, we substitute the point

(a, 9)

into this equation:

9 = 2a - 1

. Solving for

a

gives

10 = 2a

, which simplifies to

a = 5

.

Step-by-Step Solution

1

Calculate the slope of line

l

using the two given points.

The slope of line

l

is

2

.

Using the slope formula

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

with the points

(-2, -3)

and

(2, 5)

, we get

m = \frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2

.

2

Find the equation of line

k

.

The equation of line

k

is

y = 2x - 1

.

Line

k

is parallel to line

l

, so it has the same slope,

m = 2

. Its

y

-intercept is

(0, -1)

, which gives the equation

y = 2x - 1

in slope-intercept form.

3

Solve for

a

by substituting the point

(a, 9)

into the equation of line

k

.

The value of

a

is

5

.

Substituting

x = a

and

y = 9

into

y = 2x - 1

gives

9 = 2a - 1

. Solving for

a

yields

10 = 2a

, or

a = 5

.

Key Concept

Linear Functions and Graphs

Question 71Question →

In the $xy$ -plane, line $l_1$ passes through the origin and has a positive slope $m$ , where $m > 1$ . Line $l_2$ is perpendicular to line $l_1$ and has a $y$ -intercept of $(0, 10)$ . The two lines intersect at point $P$ . If the distance from point $P$ to the $y$ -axis is $4$ , what is the value of $m$ ?

Show answer & explanation

Answer: 2

Answer

2

The equation of line

l_1

is

y = mx

and the equation of line

l_2

is

y = -\frac{1}{m}x + 10

. Setting these equal gives the

x

-coordinate of their intersection as

x = \frac{10m}{m^2 + 1}

. Since the distance from the intersection point to the

y

-axis is

4

, we have

\frac{10m}{m^2 + 1} = 4

. Solving this quadratic equation yields

m = 2

or

m = \frac{1}{2}

. Given that

m > 1

, the correct value is

2

.

Step-by-Step Solution

1

Write the equations of lines

l_1

and

l_2

.

Line

l_1

has a slope of

m

and passes through

(0,0)

, so its equation is

y = mx

. Line

l_2

is perpendicular to

l_1

, so its slope is

-\frac{1}{m}

. Since its

y

-intercept is

(0,10)

, its equation is

y = -\frac{1}{m}x + 10

.

Setting up the equations of the lines allows us to find their point of intersection.

2

Find the

x

-coordinate of the intersection point

P

.

Equating the two expressions for

y

gives

mx = -\frac{1}{m}x + 10

. Multiplying both sides by

m

yields

m^2 x = -x + 10m

, which simplifies to

(m^2 + 1)x = 10m

, or

x = \frac{10m}{m^2 + 1}

.

The intersection point

P

must satisfy both equations simultaneously.

3

Solve for

m

using the distance from

P

to the

y

-axis.

The distance from

P(x,y)

to the

y

-axis is given by

|x|

. Since

m > 1

,

x

is positive, so the distance is

\frac{10m}{m^2 + 1} = 4

. This simplifies to

10m = 4m^2 + 4

, or

4m^2 - 10m + 4 = 0

. Dividing by

2

gives

2m^2 - 5m + 2 = 0

. Factoring the quadratic yields

(2m - 1)(m - 2) = 0

, giving solutions of

m = \frac{1}{2}

and

m = 2

. Since

m > 1

, the slope of

l_1

must be

2

.

Applying the given distance constraint and slope condition determines the unique value of

m

.

Key Concept

The relationship between the equations of perpendicular lines and their point of intersection in the coordinate plane.

Question 72Question →

If $3(2n + 4) = 42$ , what is the value of $n$ ?

Show answer & explanation

Answer: 5

Answer

5

To solve the linear equation

3(2n + 4) = 42

, first divide both sides of the equation by 3, which simplifies it to

2n + 4 = 14

. Next, subtract 4 from both sides of the equation to isolate the term with the variable, giving

2n = 10

. Finally, divide both sides of the equation by 2 to solve for the variable, which yields

n = 5

.

Step-by-Step Solution

1

Divide both sides of the equation by 3.

2n + 4 = 14

To simplify the equation by eliminating the outer constant multiplier.

2

Subtract 4 from both sides of the equation.

2n = 10

To isolate the variable term on one side of the equation.

3

Divide both sides of the equation by 2.

n = 5

To find the value of

n

.

Key Concept

Solving a linear equation in one variable using inverse operations.

Estimated Time:45s

Question 73Question →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $y = 2x - 7$ . If the region in the first quadrant bounded by the graph of $f$ , the $x$ -axis, and the $y$ -axis has an area of $36$ , what is the $x$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 12

Answer

The

x

-intercept of the graph of

f

is

12

.

The correct answer is

12

. Since the graph of

f

is perpendicular to the line

y = 2x - 7

, its slope is

-\frac{1}{2}

. The equation of the line is

f(x) = -\frac{1}{2}x + b

, which has a

y

-intercept of

(0, b)

and an

x

-intercept of

(2b, 0)

. In the first quadrant, these intercepts form a right triangle with the axes, having legs of length

b

and

2b

. The area of this triangle is

\frac{1}{2}(2b)(b) = b^2

. Setting the area equal to

36

gives

b^2 = 36

, so

b = 6

(since

b > 0

). The

x

-intercept is

2b = 2(6) = 12

.

Step-by-Step Solution

1

Find the slope of the perpendicular line

f

.

The slope of

f

is

-\frac{1}{2}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The given line has a slope of

2

, so the slope of

f

must be

-\frac{1}{2}

.

2

Express the

y

-intercept and

x

-intercept of

f

in terms of a single variable.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(2b, 0)

, where

b > 0

.

The equation of

f

can be written as

f(x) = -\frac{1}{2}x + b

. Setting

x = 0

gives the

y

-intercept

b

. Setting

f(x) = 0

and solving for

x

gives the

x

-intercept

2b

.

3

Set up the area equation for the triangle in the first quadrant.

The area is represented by

b^2 = 36

.

The region bounded by the graph of

f

and the coordinate axes in the first quadrant forms a right triangle with perpendicular sides of lengths

b

and

2b

. The area of this triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2b \times b = b^2

.

4

Solve for

b

and calculate the

x

-intercept.

The

x

-intercept is

12

.

Solving

b^2 = 36

with the condition

b > 0

yields

b = 6

. The

x

-intercept is

2b

, which equals

2(6) = 12

.

Key Concept

Using perpendicular slopes and intercepts of linear functions to analyze geometric areas in the coordinate plane.

Estimated Time:2m 0s

Question 74Question →

A commercial drone delivery service charges a flat rate of $15.50$ dollars per delivery plus a fuel surcharge of $1.80$ dollars per mile. During a promotional event, the service offers a $20\%$ discount off the fuel surcharge only. If the total cost for a specific delivery during this promotion was $27.02$ dollars, how many miles did the drone travel for this delivery?

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. The promotional fuel surcharge is reduced by

20\%

from the normal

1.80

dollars per mile, resulting in a rate of

1.80 \times (1 - 0.20) = 1.44

dollars per mile. The linear equation modeling the total cost for a delivery of

m

miles during the promotion is

15.50 + 1.44m = 27.02

. Subtracting

15.50

from both sides gives

1.44m = 11.52

. Dividing both sides by

1.44

yields the distance

m = 8

miles.

Step-by-Step Solution

1

Determine the discounted fuel surcharge per mile.

The discounted surcharge is

1.44

dollars per mile.

A

20\%

discount is applied to the original

1.80

dollars per mile fuel surcharge:

1.80 \times (1 - 0.20) = 1.44

.

2

Set up the linear equation representing the total promotional cost.

15.50 + 1.44m = 27.02

, where

m

represents the number of miles traveled.

The total cost is the sum of the flat rate of

15.50

dollars and the promotional fuel surcharge of

1.44

dollars per mile multiplied by the number of miles.

3

Subtract the flat rate from both sides of the equation.

1.44m = 11.52

To isolate the term with the variable

m

, subtract the flat rate of

15.50

from the total promotional cost of

27.02

.

4

Solve for the distance by division.

m = 8

Divide both sides of the equation by the promotional rate per mile,

1.44

, to find the number of miles.

Key Concept

Formulating and solving a multi-step linear equation in one variable with decimals and percentages to represent a real-world scenario.

Question 75Question →

A local municipal water utility charges a flat monthly service fee plus a constant rate for each hundred cubic feet (HCF) of water consumed. During one month, a household that consumed 12 HCF of water was charged 46.00 dollars. Another household that consumed 18 HCF of water was charged 61.00 dollars. What is the flat monthly service fee, in dollars, charged by the utility?

Show answer & explanation

Answer: 16

Answer

The flat monthly service fee is 16 dollars.

By modeling the relationship as a linear function

y = mx + b

, where

x

is the consumption in HCF and

y

is the total charge in dollars, we use the points

(12, 46)

and

(18, 61)

to find the slope

m = \frac{61 - 46}{18 - 12} = 2.5

. Substituting

m = 2.5

and the point

(12, 46)

back into the linear equation gives

46 = 2.5(12) + b

, which simplifies to

46 = 30 + b

. Solving for

b

gives 16, representing the flat service fee.

Step-by-Step Solution

1

Set up a system of points representing the linear relationship between water consumed and monthly charge.

Two coordinate points are established:

(12, 46)

and

(18, 61)

, where the first coordinate is the consumption in HCF and the second is the charge in dollars.

Since the utility charges a flat fee plus a constant rate, the relationship is linear and can be solved using coordinate points.

2

Find the constant rate per HCF (the slope of the line).

The slope

m

is calculated as

\frac{61 - 46}{18 - 12} = \frac{15}{6} = 2.5

dollars per HCF.

The slope represents the constant rate of change in total cost per HCF of water consumed.

3

Determine the flat monthly service fee (the y-intercept of the line).

Using the slope-intercept equation

y = mx + b

with point

(12, 46)

and

m = 2.5

yields

46 = 2.5(12) + b \implies 46 = 30 + b \implies b = 16

.

The y-intercept represents the cost when consumption is zero, which is the flat monthly service fee.

Key Concept

Finding the y-intercept of a linear function from two points.

Estimated Time:1m 30s

Question 76Question →

A scientist is measuring the temperature of a sample that is being heated at a constant rate. At a starting time of $0$ minutes, the temperature of the sample is $12^\circ\text{C}$ . After $8$ minutes of heating, the temperature of the sample is $44^\circ\text{C}$ . If the temperature of the sample increases linearly with time, what is the temperature of the sample, in degrees Celsius, after $15$ minutes of heating?

Show answer & explanation

Answer: 72

Answer

72

The rate of temperature change is

\frac{44 - 12}{8 - 0} = 4^\circ\text{C}

per minute. Using the slope-intercept form, the temperature

T

after

t

minutes can be represented by the function

T(t) = 4t + 12

. Substituting

15

for

t

yields

T(15) = 4(15) + 12 = 60 + 12 = 72

.

Step-by-Step Solution

1

Calculate the constant rate of temperature change (slope).

4 degrees Celsius per minute

To find how much the temperature increases each minute, divide the total change in temperature by the total change in time:

\frac{44 - 12}{8 - 0} = \frac{32}{8} = 4

.

2

Set up the linear function for temperature

T(t)

over time

t

.

T(t) = 4t + 12

The initial temperature at

t = 0

is

12^\circ\text{C}

, which represents the vertical intercept. The constant rate of change is

4^\circ\text{C}

per minute.

3

Find the temperature at

t = 15

minutes.

72

Substitute

15

for

t

in the equation:

T(15) = 4(15) + 12 = 60 + 12 = 72

.

Key Concept

Linear Functions and Rates of Change

Estimated Time:1m 0s

Question 77Question →

If $2x + 7 = 15$ , what is the value of $4x - 3$ ?

Show answer & explanation

Answer: 13

Answer

13

To find the value of the expression, first solve the linear equation for the variable. Subtracting 7 from both sides of the equation yields

2x = 8

. Dividing both sides of the equation by 2 yields

x = 4

. Substituting this value back into the expression yields

4(4) - 3 = 16 - 3 = 13

. Alternatively, since

2x = 8

, multiplying both sides by 2 gives

4x = 16

. Subtracting 3 from both sides of this relation yields

4x - 3 = 16 - 3 = 13

.

Step-by-Step Solution

1

Subtract 7 from both sides of the equation

2x = 8

To isolate the variable term on one side of the equation

2

Divide both sides by 2

x = 4

To solve for the variable

3

Substitute the value of the variable into the target expression

13

To find the final evaluated value requested by the question

Key Concept

Solving a linear equation in one variable and evaluating an expression

Question 78Question →

A chemist mixes two saline solutions. Solution A is $12\%$ salt by mass, and Solution B is $30\%$ salt by mass. The mass of Solution B used in the mixture is 20 grams more than $\frac{1}{3}$ of the mass of Solution A used. If the resulting mixture is $18\%$ salt by mass, what is the mass, in grams, of Solution A used?

Show answer & explanation

Answer: 120

Answer

The mass of Solution A used is 120 grams.

The correct answer is 120. By setting the mass of Solution A to

x

, the mass of Solution B is

\frac{1}{3}x + 20

. Equating the total salt content from both individual solutions to the total salt content of the final mixture gives the linear equation

0.12x + 0.30\left(\frac{1}{3}x + 20\right) = 0.18\left(x + \frac{1}{3}x + 20\right)

. Simplifying both sides yields

0.22x + 6 = 0.24x + 3.6

. Solving this equation gives

0.02x = 2.4

, which simplifies to

x = 120

.

Step-by-Step Solution

1

Define the variable

x

as the mass, in grams, of Solution A used in the mixture, and express the mass of Solution B in terms of

x

.

Mass of Solution A =

x

grams; Mass of Solution B =

\frac{1}{3}x + 20

grams.

To set up expressions representing the mass of each solution in the mixture.

2

Calculate the mass of salt contributed by each solution and write an expression for the total mass of salt.

Salt from Solution A =

0.12x

grams; Salt from Solution B =

0.30\left(\frac{1}{3}x + 20\right) = 0.10x + 6

grams; Total salt =

0.22x + 6

grams.

To find the total amount of salt before mixing.

3

Express the total mass of the mixture and the total salt content using the final mixture's percentage.

Total mass of mixture =

\frac{4}{3}x + 20

grams; Total salt in final mixture =

0.18\left(\frac{4}{3}x + 20\right) = 0.24x + 3.6

grams.

To write the total salt content in terms of the final mixture's concentration.

4

Equate the two expressions for the total mass of salt and solve the linear equation for

x

.

0.22x + 6 = 0.24x + 3.6 \implies 2.4 = 0.02x \implies x = 120

.

To find the mass of Solution A that satisfies the mixture conditions.

Key Concept

Setting up and solving a linear equation in one variable from a mixture word problem.

Question 79Question →

For the linear function $f$ , the table below displays selected values of $x$ and their corresponding function values $f(x)$ .

$x$	$f(x)$
$-2$	$1$
$2$	$9$
$4$	$13$

What is the value of $f(10)$ ?

Show answer & explanation

Answer: 25

Answer

The value of

f(10)

is 25.

The correct answer is 25. By using the points

(-2, 1)

and

(2, 9)

, the slope of the linear function is calculated as

m = \frac{9 - 1}{2 - (-2)} = 2

. Substituting this slope and the point

(2, 9)

into the slope-intercept form,

f(x) = 2x + b

, gives

9 = 2(2) + b

, which resolves to

b = 5

. Thus, the linear function is defined by

f(x) = 2x + 5

. Evaluating the function at

x = 10

yields

f(10) = 2(10) + 5 = 25

.

Step-by-Step Solution

1

Calculate the slope of the linear function using two coordinate pairs.

Slope

m = 2

The slope of a linear function can be determined by the formula

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

. Substituting

(-2, 1)

and

(2, 9)

yields

m = \frac{9 - 1}{2 - (-2)} = \frac{8}{4} = 2

.

2

Find the y-intercept of the function to write its equation.

f(x) = 2x + 5

Substituting the slope

m = 2

and the point

(2, 9)

into the slope-intercept form

f(x) = mx + b

gives

9 = 2(2) + b

, which simplifies to

b = 5

.

3

Evaluate the function for the input value 10.

f(10) = 25

Substituting

x = 10

into the linear function equation

f(x) = 2x + 5

yields

f(10) = 2(10) + 5 = 25

.

Key Concept

Determining a linear function from a table of values and using it to find specific outputs.

Estimated Time:1m 30s

Question 80Question →

If $\frac{5}{6}(y - 2) - \frac{1}{3}(2y - 5) = \frac{3}{2}$ , what is the value of $4y$ ?

Show answer & explanation

Answer: 36

Answer

The correct answer is 36.

To solve the equation

\frac{5}{6}(y - 2) - \frac{1}{3}(2y - 5) = \frac{3}{2}

, we first clear the fractions by multiplying the entire equation by the least common denominator, 6, which yields

5(y - 2) - 2(2y - 5) = 9

. Distributing the terms on the left side gives

5y - 10 - 4y + 10 = 9

. Combining like terms simplifies this to

y = 9

. The question asks for the value of

4y

, so we multiply

9

by

4

to get the final answer of

36

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator, which is 6.

5(y - 2) - 2(2y - 5) = 9

Multiplying by 6 eliminates the fractions, simplifying the equation.

2

Distribute the coefficients across the parentheses.

5y - 10 - 4y + 10 = 9

Expanding the terms allows like terms to be combined in the next step.

3

Combine the variable terms and constant terms on the left side of the equation.

y = 9

Combining

5y - 4y

yields

y

, and combining

-10 + 10

yields

0

, isolating the variable.

4

Multiply the value of

y

by 4.

4y = 36

The question asks for the value of

4y

rather than just

y

.

Key Concept

Solving a linear equation in one variable by clearing fractions and combining like terms.

Answer

Step-by-Step Solution

Key Concept

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