All practice questions

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

Show answer & explanation

Answer: 11

Answer

11

Evaluating the linear equation by clearing the fractions with a common denominator of 10 gives

6(2x - 4) - 5(x - 3) = 12

. Distributing terms yields

12x - 24 - 5x + 15 = 12

. Combining like terms results in

7x - 9 = 12

. Solving for

x

gives

x = 3

. Finally, substituting

x = 3

into the target expression

5x - 4

yields

5(3) - 4 = 11

.

Step-by-Step Solution

1

Multiply both sides of the equation by 10 to clear the denominators.

6(2x - 4) - 5(x - 3) = 12

The least common denominator of 5 and 2 is 10. Multiplying both sides by 10 simplifies the equation by removing the fractions.

2

Distribute the constants 6 and -5 into the parentheses.

12x - 24 - 5x + 15 = 12

Applying the distributive property removes the parentheses. Be careful to distribute the negative sign for the second term:

-5 \cdot (x - 3) = -5x + 15

.

3

Combine like terms on the left side of the equation.

7x - 9 = 12

Grouping the

x

terms (

12x - 5x = 7x

) and the constant terms (

-24 + 15 = -9

) simplifies the equation.

4

Isolate the variable term

7x

and solve for

x

.

x = 3

Adding 9 to both sides gives

7x = 21

. Dividing both sides by 7 yields

x = 3

.

5

Substitute the value of

x

into the expression

5x - 4

.

11

The question asks for the value of

5x - 4

, not just

x

, so we evaluate

5(3) - 4 = 11

.

Key Concept

Solving linear equations in one variable by clearing fractions, distributing terms, and isolating the variable.

Question 102Question →

In the system of equations below,

c

is a constant.

\begin{aligned} 2x + y &= 15 \\ cx - 2y &= 6 \end{aligned}

If the solution

(x, y)

to the system of equations above lies on the line

y = 3x

in the

xy

-plane, what is the value of the constant

c

?

Show answer & explanation

Answer: 8

Answer

The value of the constant

c

is 8.

Since the solution

(x, y)

lies on the line

y = 3x

, we can substitute

y = 3x

into the first equation,

2x + y = 15

, to get

2x + 3x = 15

, which simplifies to

5x = 15

, yielding

x = 3

. Substituting

x = 3

back into

y = 3x

gives

y = 9

. Thus, the intersection point of the system is

(3, 9)

. Substituting

x = 3

and

y = 9

into the second equation,

cx - 2y = 6

, gives

c(3) - 2(9) = 6

, or

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, which simplifies to

c = 8

.

Step-by-Step Solution

1

Substitute the equation of the line

y = 3x

into the first equation of the system,

2x + y = 15

.

2x + 3x = 15

, which simplifies to

5x = 15

. Solving for

x

gives

x = 3

.

Since the solution to the system lies on the line

y = 3x

, the coordinates of the solution must satisfy all three equations.

2

Calculate the corresponding

y

-value using the relation

y = 3x

.

y = 3(3) = 9

.

The

y

-coordinate is three times the

x

-coordinate for any point on the line

y = 3x

.

3

Substitute the point

(3, 9)

into the second equation of the system,

cx - 2y = 6

.

c(3) - 2(9) = 6

, which simplifies to

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, so

c = 8

.

Since

(3, 9)

is the solution to the system, it must satisfy the second equation as well.

Key Concept

Solving systems of linear equations using substitution and finding parameter values based on geometric constraints.

Question 103Question →

A water tank is being filled at a constant rate. The volume of water in the tank, in gallons, is a linear function of the time, in minutes, since the filling process began. The volume of water in the tank was $24$ gallons after $3$ minutes of filling, and it was $40$ gallons after $7$ minutes of filling. What was the initial volume of water, in gallons, in the tank before the filling process began?

Show answer & explanation

Answer: 12

Answer

The initial volume of water in the tank was 12 gallons.

To find the initial volume of water, we must determine the y-intercept of the linear relationship between the volume and time. First, find the rate of change (slope) using the formula m = \frac{V_2 - V_1}{t_2 - t_1}. Substituting the given values gives m = \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4. Using the slope-intercept form V(t) = mt + b, substitute one of the points, such as (3, 24), to solve for b: 24 = 4(3) + b, which simplifies to b = 12. Thus, the initial volume of water is 12 gallons.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the volume with respect to time.

4

The slope is the change in volume divided by the change in time: \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4.

2

Write the linear function V(t) = mt + b using the calculated slope, and substitute one of the given points to solve for b.

b = 12

Using the point (3, 24) and the slope m = 4, we substitute into V(t) = 4t + b to get 24 = 4(3) + b, which simplifies to 24 = 12 + b, meaning b = 12.

3

Identify the initial volume of water in the tank, which is the value of V(t) when t = 0.

12

When t = 0, V(0) = 4(0) + 12 = 12, which represents the initial volume.

Key Concept

Determining the y-intercept (initial value) of a linear function given two points.

Question 104Question →

In the system of equations below,

k

is a positive integer that is a multiple of 9.

\begin{aligned} 3x + 8y &= 213 \\ x - 3y &= k \end{aligned}

If the system has a solution

(x, y)

such that

x

and

y

are both positive integers, what is the value of

k

?

Show answer & explanation

Answer: 54

Answer

54

Substituting

x = 3y + k

into the first equation yields

17y + 3k = 213

. Solving for

k

gives

k = 71 - \frac{17}{3}y

. For

k

to be a positive integer,

y

must be a positive multiple of 3 less than 12.5. Testing the possible values for

y

(3, 6, 9, 12) yields the possible values of

k

as 54, 37, 20, and 3. Since

k

must be a multiple of 9, the correct value is 54.

Step-by-Step Solution

1

Express

x

in terms of

y

and

k

using the second equation.

x = 3y + k

Isolating

x

allows for easy substitution into the first equation to eliminate one of the variables.

2

Substitute the expression for

x

into the first equation.

17y + 3k = 213

Substituting

x = 3y + k

into

3x + 8y = 213

yields

3(3y + k) + 8y = 213

, which simplifies to

9y + 3k + 8y = 213

and then to

17y + 3k = 213

.

3

Solve for

k

in terms of

y

and analyze the divisibility constraint.

k = 71 - \frac{17}{3}y

, where

y

is a multiple of 3

For

k

to be an integer, the fractional term

\frac{17}{3}y

must resolve to an integer, requiring

y

to be divisible by 3.

4

Apply the constraint that

k

must be a positive integer (

k > 0

).

y < 12.5

, meaning

y

can be 3, 6, 9, or 12

Setting

71 - \frac{17}{3}y > 0

yields

17y < 213

, or

y < 12.53

. The positive integer multiples of 3 in this range are 3, 6, 9, and 12.

5

Find the corresponding values of

k

and select the one that is a multiple of 9.

The possible values for

k

are 54, 37, 20, and 3. The only multiple of 9 is 54.

Testing the possible values of

y

gives:

y=3 \rightarrow k=54

,

y=6 \rightarrow k=37

,

y=9 \rightarrow k=20

, and

y=12 \rightarrow k=3

. Among these, 54 is the only multiple of 9.

Key Concept

Systems of Linear Equations with Integer Constraints

Estimated Time:3m 0s

Question 105Question →

For all $x \ge 0$ , the expression $(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)$ is equivalent to $ax - b\sqrt{x} - 10$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 17

Answer

17

Expanding the expression

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

yields

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10

. Combining like terms gives

6x - 11\sqrt{x} - 10

. Comparing this result to the given equivalent form

ax - b\sqrt{x} - 10

shows that

a = 6

and

b = 11

. Therefore, the value of

a + b

is

6 + 11 = 17

.

Step-by-Step Solution

1

Multiply the binomials

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

using FOIL.

(3x^{\frac{1}{2}})(2x^{\frac{1}{2}}) - 5(3x^{\frac{1}{2}}) + 2(2x^{\frac{1}{2}}) - 10

Expanding the product allows us to write it in the standard simplified form.

2

Simplify the products and combine the like terms.

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10 = 6x - 11x^{\frac{1}{2}} - 10

This simplifies the expression so we can compare it directly to the target form.

3

Rewrite

x^{\frac{1}{2}}

as

\sqrt{x}

and compare the expression to

ax - b\sqrt{x} - 10

.

6x - 11\sqrt{x} - 10 = ax - b\sqrt{x} - 10

, which gives

a = 6

and

b = 11

.

Matching corresponding coefficients allows us to determine the values of the constants

a

and

b

.

4

Calculate the sum of

a

and

b

.

a + b = 6 + 11 = 17

The question requires finding the value of

a + b

.

Key Concept

To determine equivalence between algebraic expressions, expand the terms using the distributive property, simplify, and equate the corresponding coefficients of the like terms.

Question 106Question →

In the $xy$ -plane, a line with a positive slope $m$ passes through the point $(4, -3)$ and intersects the $x$ -axis at $(p, 0)$ and the $y$ -axis at $(0, q)$ , where $p$ and $q$ are non-zero constants. If $p + q = 5$ , what is the value of $m$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

m

is

0.5

(or

1/2

).

To find the slope

m

, write the equation of the line in point-slope form:

y - (-3) = m(x - 4)

, which simplifies to

y = mx - 4m - 3

. The

y

-intercept is found by setting

x = 0

, giving

q = -4m - 3

. The

x

-intercept is found by setting

y = 0

, giving

p = 4 + \frac{3}{m}

. Substituting these expressions into

p + q = 5

gives

4 + \frac{3}{m} - 4m - 3 = 5

, which simplifies to

-4m - 4 + \frac{3}{m} = 0

. Multiplying this equation by

-m

yields the quadratic equation

4m^2 + 4m - 3 = 0

. Factoring the quadratic gives

(2m - 1)(2m + 3) = 0

. Since the line is defined to have a positive slope, the value of

m

must be positive, which is

0.5

(or

1/2

).

Step-by-Step Solution

1

Write the general equation of a line passing through

(4, -3)

with slope

m

.

y = mx - 4m - 3

Using the point-slope form

y - y_1 = m(x - x_1)

with

(x_1, y_1) = (4, -3)

establishes the relationship between

x

,

y

, and

m

.

2

Determine the value of

q

by finding the

y

-intercept of the line.

q = -4m - 3

The

y

-intercept

(0, q)

occurs when

x = 0

.

3

Determine the value of

p

by finding the

x

-intercept of the line.

p = 4 + \frac{3}{m}

The

x

-intercept

(p, 0)

occurs when

y = 0

.

4

Substitute the expressions for

p

and

q

into the given equation

p + q = 5

.

4 + \frac{3}{m} - 4m - 3 = 5

This sets up a single equation with the variable

m

using the given relationship.

5

Simplify the equation and rewrite it in standard quadratic form.

4m^2 + 4m - 3 = 0

Multiplying the simplified expression

-4m - 4 + \frac{3}{m} = 0

by

-m

eliminates the fraction and aligns it to standard quadratic form.

6

Factor the quadratic equation to find the possible values of

m

.

m = 0.5

or

m = -1.5

Factoring

(2m - 1)(2m + 3) = 0

gives the roots of the equation.

7

Filter the results based on the positive slope constraint.

m = 0.5

The problem states that the line has a positive slope, so we must reject the negative root.

Key Concept

Using linear intercepts to solve system constraints

Alternative Method

Instead of using point-slope form, use the intercept form of a linear equation:

\frac{x}{p} + \frac{y}{q} = 1

. Since the line passes through the point

(4, -3)

, substitute

x = 4

and

y = -3

to get

\frac{4}{p} - \frac{3}{q} = 1

. Given

p + q = 5

, substitute

p = 5 - q

into the equation to get

\frac{4}{5 - q} - \frac{3}{q} = 1

. Multiply both sides by the common denominator

q(5 - q)

to obtain

4q - 3(5 - q) = q(5 - q) \implies 7q - 15 = 5q - q^2 \implies q^2 + 2q - 15 = 0

. Factoring gives

(q + 5)(q - 3) = 0

. This yields

q = -5

or

q = 3

. If

q = 3

, then

p = 2

, and the slope is

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

(negative). If

q = -5

, then

p = 10

, and the slope is

m = -\frac{q}{p} = -\frac{-5}{10} = 0.5

(positive). Since the slope must be positive, the slope of the line is

0.5

.

Estimated Time:2m 30s

Question 107Question →

In a certain video game, a player earns 15 points for completing a level and loses 3 points for each hint they use. If a player wants to score at least 6 points on a level, what is the maximum number of hints they can use?

Show answer & explanation

Answer: 3

Answer

The maximum number of hints the player can use is 3.

To find the maximum number of hints, we construct the inequality representing the player's score:

15 - 3h \geq 6

, where

h

is the number of hints used. Subtracting 15 from both sides gives

-3h \geq -9

. Dividing both sides by

-3

and reversing the inequality sign yields

h \leq 3

. This means the player can use at most 3 hints to achieve a score of at least 6 points. Thus, the maximum number of hints is 3.

Step-by-Step Solution

1

Set up the inequality representing the score requirement.

15 - 3h \geq 6

The starting score is 15, and 3 points are lost for each hint

h

. The final score must be at least (greater than or equal to) 6.

2

Subtract 15 from both sides of the inequality.

-3h \geq -9

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

3

Divide both sides by -3 and flip the inequality sign.

h \leq 3

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Solving multi-step linear inequalities with negative coefficients

Estimated Time:45s

Question 108Question →

A local store sells two types of coffee beans: Arabica and Robusta. On Monday, the store sold $8$ pounds of Arabica coffee and $5$ pounds of Robusta coffee for a total of $\$62.00$ . On Tuesday, the store sold $4$ pounds of Arabica coffee and $7$ pounds of Robusta coffee for a total of $\$58.00$ . What is the cost, in dollars, of one pound of Robusta coffee?

Show answer & explanation

Answer: 6

Answer

The cost of one pound of Robusta coffee is 6 dollars.

The correct answer is 6. By setting up the system of equations

8a + 5r = 62

and

4a + 7r = 58

, we can multiply the second equation by 2 to get

8a + 14r = 116

. Subtracting the first equation from this yields

9r = 54

, which simplifies to

r = 6

.

Step-by-Step Solution

1

Define variables for the price per pound of Arabica coffee (

a

) and Robusta coffee (

r

), then set up a system of linear equations based on the given store sales information.

Equation 1:

8a + 5r = 62

Equation 2:

4a + 7r = 58

This translates the word problem context into a solvable system of mathematical equations.

2

Multiply the second equation by

2

to make the coefficients of

a

equal in both equations.

8a + 14r = 116

This prepares the system for solving by the elimination method.

3

Subtract the first equation (

8a + 5r = 62

) from the new equation (

8a + 14r = 116

) to eliminate

a

and solve for

r

.

9r = 54

, which simplifies to

r = 6

Subtracting the equations eliminates one variable, leaving a single-variable equation that is easily solved.

Key Concept

Solving systems of two linear equations in two variables using the elimination method.

Estimated Time:1m 30s

Question 109Question →

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are positive constants. The line $y = -2x$ is perpendicular to the line that passes through the origin $(0,0)$ and the midpoint of the segment connecting the two intercepts of $f$ . If $r = 12$ , what is the value of $s$ ?

Show answer & explanation

Answer: 24

Answer

24

The midpoint of the segment connecting

(0, 12)

and

(s, 0)

is

(\frac{s}{2}, 6)

. The line passing through the origin

(0, 0)

and this midpoint has a slope of

\frac{6}{s/2} = \frac{12}{s}

. Since this line is perpendicular to the line

y = -2x

, which has a slope of

-2

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. Equating these two slopes, we get

\frac{12}{s} = \frac{1}{2}

, which simplifies to

s = 24

.

Step-by-Step Solution

1

Find the coordinates of the intercepts and their midpoint.

The intercepts are

(0, 12)

and

(s, 0)

, and their midpoint is

(\frac{s}{2}, 6)

.

The intercepts of the function

f

form a line segment whose midpoint must be calculated.

2

Calculate the slope of the line passing through the origin and the midpoint.

The slope is

\frac{12}{s}

.

A line passing through the origin

(0,0)

and a point

(x_1, y_1)

has a slope of

\frac{y_1}{x_1}

.

3

Relate the slope of the line to the perpendicular line

y = -2x

.

The slope of the line must be

\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other, and the negative reciprocal of

-2

is

\frac{1}{2}

.

4

Solve for the value of

s

.

s = 24

.

Setting the two expressions for the slope equal,

\frac{12}{s} = \frac{1}{2}

, yields

s = 24

.

Key Concept

Linear Functions and Graphs

Estimated Time:2m 30s

Question 110Question →

A student wants to buy $x$ notebooks and $y$ pens. The student must buy at least $3$ notebooks. Each notebook costs $3$ dollars and each pen costs $2$ dollars. If the student can spend a maximum of $15$ dollars, what is the maximum number of pens the student can buy?

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. The constraints are represented by the system of inequalities

x \geq 3

and

3x + 2y \leq 15

. To find the maximum number of pens

y

, we minimize the number of notebooks

x

by setting

x = 3

. Substituting

x = 3

into the budget inequality gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

. Subtracting

9

from both sides results in

2y \leq 6

. Dividing by

2

yields

y \leq 3

. Therefore, the maximum number of pens the student can buy is 3.

Step-by-Step Solution

1

Set up the system of inequalities from the given constraints.

The system of inequalities is

x \geq 3

and

3x + 2y \leq 15

, where

x

represents the number of notebooks and

y

represents the number of pens.

To represent the minimum number of notebooks required and the total budget limit.

2

Substitute the minimum possible value of

x

into the budget inequality to maximize

y

.

Setting

x = 3

gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

.

Since the cost per notebook is positive, minimizing the number of notebooks leaves the maximum amount of budget for purchasing pens.

3

Solve the inequality for

y

.

2y \leq 6 \implies y \leq 3

.

To find the upper bound for the number of pens the student can buy.

Key Concept

Solving a system of linear inequalities in a real-world context to find a maximum value.

Question 111Question →

In the $xy$ -plane, a point $(x, y)$ is a solution to the system of inequalities below.

y \leq -3x + 14

y \leq 4x - 7

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 5

Answer

The maximum possible value of

y

is 5.

The maximum value of

y

is found at the intersection of the two boundary lines,

y = -3x + 14

and

y = 4x - 7

. Solving the system of equations gives

x = 3

and

y = 5

. Because the inequalities are both of the 'less than or equal to' type, any point in the solution set must lie on or below both lines. For

x < 3

, the value of

y

is limited by

y \leq 4x - 7

, which is strictly less than 5. For

x > 3

, the value of

y

is limited by

y \leq -3x + 14

, which is also strictly less than 5. Thus, the maximum value of

y

is 5.

Step-by-Step Solution

1

Set the two boundary equations equal to find their intersection point.

-3x + 14 = 4x - 7

The maximum value of

y

under these constraints occurs at the intersection of the two boundary lines.

2

Solve the equation for

x

.

x = 3

Isolating

x

determines the x-coordinate of the intersection point.

3

Substitute

x = 3

back into one of the boundary equations to solve for

y

.

y = 5

This yields the y-coordinate of the intersection point, which is the upper bound for

y

in the solution set.

Key Concept

Determining the maximum value of a variable in the solution set of a system of linear inequalities by finding the intersection of their boundary lines.

Estimated Time:1m 30s

Question 112Question →

The table below shows some values of the linear function $h$ .

$x$	$h(x)$
$2$	$k - 4$
$5$	$k + 8$
$8$	$2k + 2$

If $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

The value of the constant

k

is

18

.

Since the function

h

is linear, its rate of change (slope) is constant. Calculating the slope using the first two coordinate pairs

(2, k - 4)

and

(5, k + 8)

gives

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{12}{3} = 4

. Using the next two coordinate pairs

(5, k + 8)

and

(8, 2k + 2)

gives the slope as

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{k - 6}{3}

. Setting these two slope values equal to each other gives the equation

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

. Multiplying both sides by

3

results in

k - 6 = 12

, and adding

6

to both sides yields

k = 18

.

Step-by-Step Solution

1

Identify that the rate of change (slope) of a linear function is constant between any two points.

The slope calculated from the first two points must equal the slope calculated from the second and third points.

This relationship allows us to set up an algebraic equation to solve for the unknown constant

k

.

2

Calculate the slope using the first two points:

(2, k - 4)

and

(5, k + 8)

.

Slope =

\frac{(k + 8) - (k - 4)}{5 - 2} = \frac{k + 8 - k + 4}{3} = \frac{12}{3} = 4

.

This simplifies to a constant numerical value of

4

for the slope of the function.

3

Calculate the slope using the second and third points:

(5, k + 8)

and

(8, 2k + 2)

.

Slope =

\frac{(2k + 2) - (k + 8)}{8 - 5} = \frac{2k + 2 - k - 8}{3} = \frac{k - 6}{3}

.

This provides a second expression for the slope in terms of the variable

k

.

4

Equate the two slope expressions and solve for

k

.

\frac{k - 6}{3} = 4 \implies k - 6 = 12 \implies k = 18

.

Setting the two expressions equal and solving isolating

k

gives the correct value of

18

.

Key Concept

A linear function has a constant rate of change (slope) between any two points on its graph.

Question 113Question →

A local delivery service charges a flat fee of $12$ dollars plus $1.50$ dollars per mile to deliver a package. If a customer wants to spend no more than $30$ dollars for a package delivery, what is the maximum number of miles the delivery service can travel?

Show answer & explanation

Answer: 12

Answer

The maximum number of miles the delivery service can travel is 12.

The total delivery cost is the sum of the flat fee (

12

dollars) and the rate per mile (

1.50

dollars multiplied by

m

miles), which is represented by

1.50m + 12

. Since the total cost cannot exceed

30

dollars, the inequality is

1.50m + 12 \leq 30

. Subtracting

12

from both sides gives

1.50m \leq 18

. Dividing both sides by

1.50

yields

m \leq 12

. Thus, the maximum distance the delivery service can travel is

12

miles.

Step-by-Step Solution

1

Set up the inequality representing the delivery cost constraint.

1.50m + 12 \leq 30

The total cost of the delivery is the flat fee of

12

dollars plus

1.50

dollars per mile,

m

, which must be less than or equal to the budget of

30

dollars.

2

Subtract 12 from both sides of the inequality.

1.50m \leq 18

To isolate the variable term, subtract the constant flat fee from both sides of the inequality.

3

Divide both sides of the inequality by 1.50.

m \leq 12

Dividing by the per-mile rate calculates the maximum distance constraint on the variable

m

.

Key Concept

Setting up and solving a one-variable linear inequality to determine a maximum boundary value in context.

Question 114Question →

In the $xy$ -plane, a system of two linear equations has no solutions. One of the equations in the system is $4x - 6y = 15$ . The graph of the second equation is a line that passes through the points $(3, k)$ and $(9, 7)$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of

k

is

3

.

For a system of linear equations to have no solutions, the lines representing the equations must be parallel, which means they have equal slopes but different y-intercepts. The first equation,

4x - 6y = 15

, can be rewritten in slope-intercept form as

y = \frac{2}{3}x - 2.5

, showing its slope is

\frac{2}{3}

. The slope of the second line, passing through

(3, k)

and

(9, 7)

, is given by

\frac{7 - k}{9 - 3} = \frac{7 - k}{6}

. Setting the two slopes equal yields

\frac{2}{3} = \frac{7 - k}{6}

. Multiplying both sides by

6

gives

4 = 7 - k

, which solves to

k = 3

. Substituting

k = 3

back into the second line gives

y = \frac{2}{3}x + 1

. Since the slopes are equal and the y-intercepts (

-2.5

and

1

) are different, the lines are parallel and distinct, confirming there are no solutions.

Step-by-Step Solution

1

Find the slope of the first line by converting the equation to slope-intercept form.

Slope is

\frac{2}{3}

and y-intercept is

-2.5

.

Converting

4x - 6y = 15

to

y = \frac{2}{3}x - 2.5

reveals the slope of the first line.

2

Express the slope of the second line using the coordinates of the two points on the line.

Slope expression is

\frac{7 - k}{6}

.

Applying the slope formula

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

to

(3, k)

and

(9, 7)

defines the slope in terms of

k

.

3

Equate the two slopes and solve for the unknown parameter.

k = 3

Since the system has no solutions, the lines must be parallel and have equal slopes. Setting

\frac{2}{3} = \frac{7 - k}{6}

and solving gives

k = 3

.

Key Concept

For a system of two linear equations to have no solutions, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts.

Estimated Time:2m 0s

Question 115Question →

A closed triangular region in the coordinate plane is defined by the following system of linear inequalities:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ x &\geq 1 \end{aligned}

What is the maximum possible value of the expression $2x + y$ for any point $(x, y)$ that lies within or on the boundary of this region?

Show answer & explanation

Answer: 12

Answer

The maximum possible value of the expression

2x + y

is 12.

To find the maximum possible value of the expression

2x + y

subject to the given system of inequalities, we first identify the boundary lines and find the vertices of the bounded triangular region in the coordinate plane. The boundary lines are

y = 2x - 4

,

y = -x + 8

, and

x = 1

. The intersection of

y = 2x - 4

and

y = -x + 8

occurs at

x = 4, y = 4

, which gives vertex

(4, 4)

. The intersection of

y = 2x - 4

and

x = 1

occurs at

(1, -2)

. The intersection of

y = -x + 8

and

x = 1

occurs at

(1, 7)

. Evaluating the linear expression

2x + y

at these three vertices gives

2(4) + 4 = 12

,

2(1) - 2 = 0

, and

2(1) + 7 = 9

. By the corner point theorem, the maximum value of a linear function on a closed polygonal region occurs at one of the vertices. Comparing the values, the maximum possible value is 12.

Step-by-Step Solution

1

Find the vertex formed by the intersection of the boundary lines

y = 2x - 4

and

y = -x + 8

.

Vertex

A(4, 4)

Setting the two equations equal:

2x - 4 = -x + 8 \implies 3x = 12 \implies x = 4

. Substituting

x = 4

back into either equation yields

y = 4

.

2

Find the vertex formed by the intersection of the boundary line

y = 2x - 4

and the vertical line

x = 1

.

Vertex

B(1, -2)

Substituting

x = 1

into

y = 2x - 4

gives

y = 2(1) - 4 = -2

.

3

Find the vertex formed by the intersection of the boundary line

y = -x + 8

and the vertical line

x = 1

.

Vertex

C(1, 7)

Substituting

x = 1

into

y = -x + 8

gives

y = -1 + 8 = 7

.

4

Evaluate the expression

2x + y

at each of the three vertices.

At

A(4, 4)

, the value is

12

; at

B(1, -2)

, the value is

0

; at

C(1, 7)

, the value is

9

.

According to the Corner Point Theorem of linear programming, the maximum or minimum of a linear objective function on a closed bounded region must occur at one of the vertices.

5

Identify the maximum value from the evaluated points.

The maximum value is 12.

Comparing the values 12, 0, and 9 shows that 12 is the largest value.

Key Concept

Linear Programming and Systems of Inequalities

Question 116Question →

An online streaming service offers the two monthly subscription plans described in the table below:

Plan	Monthly fee	Cost per premium movie rental
Plan A	$12	$1.50
Plan B	$27 (includes first 4 rentals)	$0.75 (for each rental after the first 4)

If a user rented $m$ premium movies in a month, where $m > 4$ , and the total cost for both plans would be the same, what is the value of $m$ ?

Show answer & explanation

Answer: 16

Answer

The value of

m

that results in the same total monthly cost for both plans is 16.

The correct answer of 16 represents the exact number of premium movie rentals where the total monthly cost for both plans is equal to $36. Any other number of movie rentals will result in different costs for the two plans.

Step-by-Step Solution

1

Define the cost equation for Plan A

\text{Cost}_A = 12 + 1.50m

Plan A charges a flat

12 monthly fee plus

1.50 for each of the

m

movies rented.

2

Define the cost equation for Plan B

\text{Cost}_B = 27 + 0.75(m - 4)

Plan B charges a

27 monthly fee that covers the first 4 movies, and

0.75 for each of the

m - 4

additional movies rented because

m > 4

.

3

Equate the two cost expressions and solve for

m

12 + 1.50m = 27 + 0.75(m - 4) \implies 12 + 1.50m = 24 + 0.75m \implies 0.75m = 12 \implies m = 16

To find when the costs are identical, set the two algebraic expressions equal to each other and isolate the variable

m

.

Key Concept

Setting up and solving linear equations in one variable from context

Alternative Method

Instead of setting up full equations, we can look at the cost difference at

m = 4

. At

4

movies, Plan A costs

12 + 1.50(4) = 18

dollars, and Plan B costs

27

dollars (since

4

movies are included). The price difference is

27 - 18 = 9

dollars. For each movie rented beyond

4

, the cost of Plan A increases by

1.50

dollars while Plan B only increases by

0.75

dollars. The rate of change difference is

1.50 - 0.75 = 0.75

dollars per movie. To bridge the initial

9

dollar difference, the user needs to rent

\frac{9}{0.75} = 12

more movies. Thus, the total number of movies is

4 + 12 = 16

.

Estimated Time:1m 30s

Question 117Question →

In the $xy$ -plane, a line passes through the points $(2, 9)$ and $(5, 21)$ . What is the slope of this line?

Show answer & explanation

Answer: 4

Answer

The slope of the line is 4.

The slope

m

of a line passing through the points

(x_1, y_1)

and

(x_2, y_2)

is determined by the formula

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

. Substituting the given points

(2, 9)

and

(5, 21)

into the formula gives

m = \frac{21 - 9}{5 - 2} = \frac{12}{3} = 4

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (2, 9)

and

(x_2, y_2) = (5, 21)

These points are used in the slope formula.

2

Apply the slope formula

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

to set up the calculation.

m = \frac{21 - 9}{5 - 2}

The slope is defined as the change in

y

divided by the change in

x

.

3

Simplify the fraction to calculate the final slope.

m = \frac{12}{3} = 4

Dividing the vertical change by the horizontal change yields the slope.

Key Concept

Calculating the slope of a line given two points in the coordinate plane.

Question 118Question →

The linear function $f$ is defined such that its graph in the $xy$ -plane is perpendicular to the line $3x - 4y = 20$ . The graph of another linear function, $g$ , is the result of shifting the graph of $f$ right by $5$ units and down by $4$ units. If $f(0) = 8$ and the graph of $g$ intersects the $x$ -axis at the point $(k, 0)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

The slope of the line

3x - 4y = 20

is

\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal,

-\frac{4}{3}

. With a

y

-intercept of

(0, 8)

, the function is

f(x) = -\frac{4}{3}x + 8

. Translating this function

5

units right and

4

units down yields

g(x) = f(x - 5) - 4 = -\frac{4}{3}(x - 5) + 4 = -\frac{4}{3}x + \frac{32}{3}

. Setting

g(k) = 0

to find the

x

-intercept gives

-\frac{4}{3}k + \frac{32}{3} = 0

, which solves to

k = 8

.

Step-by-Step Solution

1

Find the slope of the line

3x - 4y = 20

.

The slope is

\frac{3}{4}

.

Converting the line equation to slope-intercept form

y = mx + b

gives

y = \frac{3}{4}x - 5

, where the coefficient of

x

represents the slope.

2

Find the slope of

f

.

The slope of

f

is

-\frac{4}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the function

f(x)

.

f(x) = -\frac{4}{3}x + 8

.

Since the

y

-intercept is given by

f(0) = 8

and the slope is

-\frac{4}{3}

, the slope-intercept form is

f(x) = -\frac{4}{3}x + 8

.

4

Determine the function

g(x)

by translating

f(x)

5

units right and

4

units down.

g(x) = -\frac{4}{3}x + \frac{32}{3}

.

A translation of

f(x)

right by

5

units and down by

4

units corresponds to

g(x) = f(x - 5) - 4

. Substituting

x-5

into

f(x)

gives

g(x) = -\frac{4}{3}(x - 5) + 8 - 4 = -\frac{4}{3}x + \frac{32}{3}

.

5

Set

g(k) = 0

and solve for

k

.

k = 8

.

The graph of

g

intersects the

x

-axis at

(k, 0)

, which means

g(k) = 0

. Solving

-\frac{4}{3}k + \frac{32}{3} = 0

yields

k = 8

.

Key Concept

Writing linear functions using perpendicular slopes and performing vertical and horizontal transformations.

Question 119Question →

In the

xy

-plane, a system of inequalities is defined as follows:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ y &\geq \frac{1}{2}x \end{aligned}

How many points

(x, y)

with integer coordinates satisfy this system of inequalities?

Show answer & explanation

Answer: 5

Answer

The total number of points with integer coordinates that satisfy the system is 5.

The system of inequalities defines a closed triangular region in the coordinate plane. Finding the vertices of this triangle gives the horizontal boundaries for

x

, which are

\frac{8}{3} \approx 2.67

and

\frac{16}{3} \approx 5.33

. The only integers in this range are

x = 3

,

x = 4

, and

x = 5

. Testing each of these integers in the inequalities shows that when

x = 3

,

y

can be

2, 3, 4,

or

5

(4 points); when

x = 4

,

y

can only be

4

(1 point); and when

x = 5

, there are no valid solutions. Adding these together gives a total of 5 points.

Step-by-Step Solution

1

Find the intersection points of the three boundary lines to define the vertices of the solution region.

The intersection of

y = 2x - 4

and

y = \frac{1}{2}x

is at

x = \frac{8}{3} \approx 2.67, y = \frac{4}{3} \approx 1.33

.
The intersection of

y = -x + 8

and

y = \frac{1}{2}x

is at

x = \frac{16}{3} \approx 5.33, y = \frac{8}{3} \approx 2.67

.
The intersection of

y = 2x - 4

and

y = -x + 8

is at

x = 4, y = 4

.

Finding the vertices determines the exact boundaries of the solution region in the coordinate plane.

2

Identify the integer

x

-coordinates that lie within the horizontal span of the region.

The

x

-coordinates of the region range from

\frac{8}{3} \approx 2.67

to

\frac{16}{3} \approx 5.33

. The integers in this interval are

x = 3

,

x = 4

, and

x = 5

.

Since both coordinates must be integers, we only need to test integer values of

x

that fall within the boundaries of the region.

3

Find the integer

y

-values for each candidate

x

-value.

For

x = 3

: The inequalities require

y \geq 2(3) - 4 = 2

,

y \leq -3 + 8 = 5

, and

y \geq \frac{1}{2}(3) = 1.5

. Thus,

2 \leq y \leq 5

. The integer solutions are

y = 2, 3, 4, 5

(4 points).
For

x = 4

: The inequalities require

y \geq 2(4) - 4 = 4

,

y \leq -4 + 8 = 4

, and

y \geq \frac{1}{2}(4) = 2

. Thus,

4 \leq y \leq 4

, which means

y = 4

(1 point).
For

x = 5

: The inequalities require

y \geq 2(5) - 4 = 6

and

y \leq -5 + 8 = 3

. No real number

y

can satisfy both

y \geq 6

and

y \leq 3

(0 points).

Evaluating the system at each candidate

x

-value reveals the set of matching integer

y

-values.

4

Sum the number of points found for each integer

x

-value.

4 points (when

x = 3

) + 1 point (when

x = 4

) + 0 points (when

x = 5

) = 5 points.

This yields the total number of integer coordinate pairs

(x, y)

that satisfy the system of inequalities.

Key Concept

Analyzing a bounded region defined by a system of linear inequalities to find discrete integer solutions (lattice points).

Alternative Method

Graph the three boundary lines on a grid:

y = 2x - 4

(solid line, shaded above),

y = -x + 8

(solid line, shaded below), and

y = \frac{1}{2}x

(solid line, shaded above). Identify the triangular intersection region on the grid and count the grid intersections (lattice points) that lie within or on the boundaries of this shaded triangle.

Estimated Time:2m 30s

Question 120Question →

In the system of equations below, $a$ and $b$ are constants.

\begin{aligned} ax + by &= 24 \\ 2x - 5y &= -7 \end{aligned}

If the system has the same unique solution $(x, y)$ for all values of $a$ and $b$ that satisfy the equation $4a + 3b = 24$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x + y

is

7

.

The correct answer is

7

. The solution to the system must satisfy

ax + by = 24

for any constants

a

and

b

that satisfy the constraint

4a + 3b = 24

. By matching the coefficients of

a

and

b

in both equations, we find

x = 4

and

y = 3

. To verify, we substitute these coordinates into the second equation:

2(4) - 5(3) = 8 - 15 = -7

, which is correct. The sum of the coordinates is

x + y = 4 + 3 = 7

.

Step-by-Step Solution

1

Analyze the constraint on the constants

a

and

b

given by

4a + 3b = 24

and relate it to the first equation

ax + by = 24

.

Since the first equation

ax + by = 24

must hold for all pairs of

(a, b)

satisfying

4a + 3b = 24

, the coefficients must correspond directly, meaning

x = 4

and

y = 3

.

This shows that

(4, 3)

is a point that lies on the line

ax + by = 24

regardless of the specific values of

a

and

b

, as long as they satisfy the given constraint.

2

Prove the uniqueness of the point

(4, 3)

by testing specific values for

(a, b)

that satisfy the constraint

4a + 3b = 24

.

If

b = 0

, then

4a = 24 \implies a = 6

. The equation becomes

6x = 24 \implies x = 4

. If

a = 0

, then

3b = 24 \implies b = 8

. The equation becomes

8y = 24 \implies y = 3

. This confirms

(x, y) = (4, 3)

is the unique point.

Choosing convenient test values for

a

and

b

helps isolate the variables

x

and

y

to verify the coordinates of the solution.

3

Substitute the point

(4, 3)

into the second equation of the system,

2x - 5y = -7

, to verify that it is consistent.

2(4) - 5(3) = 8 - 15 = -7

. Since this statement is true,

(4, 3)

is the consistent unique solution to the system of equations.

A solution to a system of equations must satisfy all equations in the system.

4

Find the requested value of

x + y

using the coordinates of the solution.

x + y = 4 + 3 = 7

.

The question asks for the sum of the coordinates of the solution.

Key Concept

Systems of linear equations with parameter constraints

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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