Algebra Practice Questions — GRE General Test — Page 15

Question 281Question →

An investor divided a total capital of $\$20,000$ between two accounts. Account A pays a simple annual interest rate of $6\%$ , and Account B pays a simple annual interest rate of $9\%$ . If the total interest earned from both accounts at the end of one year was $\$1,470$ , how much money was invested in Account A?

\$8,000

\$9,000

\$9,800

\$11,000

\$12,000

Show answer & explanation

Answer: $\$11,000$

Answer

The amount invested in Account A was

\$11,000

.

Let

x

represent the amount invested in Account A. The remaining capital,

20,000 - x

, is invested in Account B. The total interest earned in one year is given by

0.06x + 0.09(20,000 - x) = 1,470

. Expanding gives

0.06x + 1,800 - 0.09x = 1,470

, which simplifies to

-0.03x = -330

. Dividing by

-0.03

yields

x = 11,000

. Thus,

\$11,000

was invested in Account A.

Step-by-Step Solution

1

Define the variable for the unknown quantity.

Let

x

be the amount in dollars invested in Account A. Then

(20,000 - x)

is the amount invested in Account B.

Expressing both quantities in terms of a single variable allows setting up a single linear equation.

2

Formulate the total interest equation.

0.06x + 0.09(20,000 - x) = 1,470

Total interest is the sum of interest from Account A (

6\%

of

x

) and Account B (

9\%

of

20,000 - x

).

3

Expand and simplify the algebraic equation.

0.06x + 1,800 - 0.09x = 1,470 \implies -0.03x + 1,800 = 1,470

Distribute

0.09

across

(20,000 - x)

and combine like terms.

4

Isolate the variable

x

.

-0.03x = 1,470 - 1,800 \implies -0.03x = -330 \implies x = 11,000

Subtract

1,800

from both sides and divide by

-0.03

to find

x

.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 282Question →

A coffee merchant creates a $50$ -pound custom mixture combining Grade A coffee beans, which cost $\$8$ per pound, and Grade B coffee beans, which cost $\$14$ per pound. If the total cost of the mixture must be at least $\$460$ and at most $\$520$ , which of the following could be the weight, in pounds, of Grade A coffee beans used in the mixture? Select all such weights.

Select all that apply

25

32

35

40

45

Show answer & explanation

Answer: $32$ ; $35$ ; $40$

Answer

The weight of Grade A coffee beans can be 32 pounds, 35 pounds, or 40 pounds.

Let

x

represent the number of pounds of Grade A coffee beans. The remaining

50 - x

pounds consist of Grade B beans. The total cost of the mixture is given by

8x + 14(50 - x) = 700 - 6x

. Setting up the given inequality constraints, we have

460 \le 700 - 6x \le 520

. Subtracting 700 yields

-240 \le -6x \le -180

. Dividing by

-6

and reversing the inequality signs gives

30 \le x \le 40

. Therefore, any weight of Grade A beans between 30 and 40 pounds inclusive is valid. The values 32, 35, and 40 satisfy this condition.

Step-by-Step Solution

1

Define variables for the quantities of Grade A and Grade B beans.

Let

x

be the weight in pounds of Grade A beans. Then

50 - x

is the weight in pounds of Grade B beans.

The total weight of the mixture is fixed at 50 pounds.

2

Set up an expression for the total cost of the mixture in terms of

x

.

Total Cost =

8x + 14(50 - x) = 700 - 6x

.

Multiply the weight of each component by its price per pound.

3

Set up the compound inequality representing the given cost constraints.

460 \le 700 - 6x \le 520

.

The total cost must be at least

\$460

and at most

\$520

.

4

Solve the compound inequality for

x

.

Subtracting 700 from all parts gives

-240 \le -6x \le -180

. Dividing by

-6

and reversing the inequality signs yields

30 \le x \le 40

.

Dividing by a negative number reverses the direction of the inequality signs.

5

Identify which options fall within the valid range

[30, 40]

.

The values 32, 35, and 40 fall within the range

30 \le x \le 40

.

Any weight between 30 and 40 pounds inclusive satisfies the cost constraint.

Key Concept

Linear Modeling and Inequality Constraints in Mixture Problems

Estimated Time:1m 40s

Question 283Question →

A plumbing service charges a flat diagnostic fee of $\$45$ plus $\$65$ for each hour of repair work. If the total bill for a repair job was $\$305$ , how many hours of repair work were performed?

Show answer & explanation

Answer: 4

Answer

The total number of hours of repair work performed was 4.

To determine the number of repair hours, represent the scenario with the linear equation

45 + 65h = 305

, where

h

is the number of hours. Subtracting 45 from both sides yields

65h = 260

. Dividing 260 by 65 gives

h = 4

.

Step-by-Step Solution

1

Formulate the linear equation from the word problem context.

45 + 65h = 305

The total charge consists of a one-time fixed fee of

45 plus a variable fee of

65 per hour

h

.

2

Isolate the variable term by subtracting 45 from both sides of the equation.

65h = 260

Subtracting the constant fee isolates the total cost incurred from hourly work.

3

Divide both sides by the coefficient of the variable to solve for

h

.

h = 4

Dividing the total labor charge (

260) by the hourly rate (

65) yields the number of hours worked.

Key Concept

Linear Equations in One Variable

Question 284Question →

In the $xy$ -plane, line $L$ is given by the equation $4x + 3y = 24$ . Line $M$ passes through the origin $(0,0)$ and is perpendicular to line $L$ . Which of the following statements must be true? Select all that apply.

Select all that apply

Line

M

passes through the point

(8, 6)

.

The point of intersection of line

L

and line

M

lies in Quadrant I.

The slope of line

M

is

-\frac{3}{4}

.

The area of the triangular region bounded by line

L

and the coordinate axes is

48

square units.

The

x

-intercept of line

L

is

(0, 8)

.

Show answer & explanation

Answer: Line $M$ passes through the point $(8, 6)$ .; The point of intersection of line $L$ and line $M$ lies in Quadrant I.

Answer

The statements confirming that line

M

passes through

(8, 6)

and that the intersection point of lines

L

and

M

lies in Quadrant I are correct.

The correct statements accurately calculate the negative reciprocal slope of line

M

as

\frac{3}{4}

, verify that

(8,6)

satisfies

y = \frac{3}{4}x

, and correctly find that the intersection of lines

L

and

M

has positive

x

and

y

coordinates in Quadrant I.

Step-by-Step Solution

1

Find the slope and equation of line

L

.

Line

L

in slope-intercept form is

y = -\frac{4}{3}x + 8

, with slope

m_L = -\frac{4}{3}

,

x

-intercept

(6,0)

, and

y

-intercept

(0,8)

.

Converting to slope-intercept form

y = mx + b

exposes the slope and intercepts of line

L

.

2

Determine the slope and equation of perpendicular line

M

.

Slope

m_M = -\frac{1}{m_L} = \frac{3}{4}

. Line

M

passes through the origin, so its equation is

y = \frac{3}{4}x

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

3

Evaluate the statement regarding point

(8,6)

on line

M

.

y = \frac{3}{4}(8) = 6

. The point

(8,6)

satisfies the equation of line

M

.

Substituting the coordinates into the line equation verifies point membership.

4

Find the intersection point of line

L

and line

M

.

Substituting

y = \frac{3}{4}x

into

4x + 3y = 24

yields

4x + 3\left(\frac{3}{4}x\right) = 24 \implies \frac{25}{4}x = 24 \implies x = \frac{96}{25}

. Then

y = \frac{72}{25}

.

Both

x

and

y

coordinates are positive, which confirms the intersection lies in Quadrant I.

Key Concept

Perpendicular Line Slopes and Intersections in Coordinate Geometry

Question 285Question →

A company has a total of $150$ employees assigned to either the Marketing department or the Development department. The number of employees in the Development department is $30$ more than $3$ times the number of employees in the Marketing department. How many employees work in the Marketing department?

25

30

40

45

120

Show answer & explanation

Answer: 30

Answer

The number of employees working in the Marketing department is 30.

Letting

m

represent the Marketing department employees, the Development department has

3m + 30

employees. Summing both yields

m + 3m + 30 = 150

, which simplifies to

4m + 30 = 150

. Subtracting 30 gives

4m = 120

, and dividing by 4 yields

m = 30

.

Step-by-Step Solution

1

Define variables for each department.

Let

m

be the number of employees in Marketing. Then the number of employees in Development is

3m + 30

.

Expressing both quantities in terms of a single variable simplifies setting up a linear equation.

2

Set up the linear equation representing total employees.

m + (3m + 30) = 150

The sum of employees in both departments equals the company total of 150.

3

Combine like terms.

4m + 30 = 150

Adding

m

and

3m

gives

4m

.

4

Isolate the variable term and solve for

m

.

4m = 120 \implies m = 30

Subtract 30 from both sides and then divide by 4.

Key Concept

Linear Equations in One Variable - Word Problem Modeling

Estimated Time:1m 0s

Question 286Question →

In the $xy$ -plane, line $L$ passes through the points $(-3, 4)$ and $(5, -2)$ . Line $M$ is perpendicular to line $L$ and passes through the point $(1, -1)$ . What is the $y$ -intercept of line $M$ ?

-\frac{7}{3}

-\frac{7}{4}

-\frac{1}{4}

\frac{7}{3}

\frac{1}{3}

Show answer & explanation

Answer: $-\frac{7}{3}$

Answer

The

y

-intercept of line

M

is

-\frac{7}{3}

.

The slope of line

L

is

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

. The perpendicular line

M

has slope

m_M = \frac{4}{3}

. Substituting point

(1, -1)

into the line equation gives

y - (-1) = \frac{4}{3}(x - 1)

, which simplifies to

y = \frac{4}{3}x - \frac{7}{3}

. Thus, the

y

-intercept is

-\frac{7}{3}

.

Step-by-Step Solution

1

Calculate the slope of line

L

using the slope formula

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

.

m_L = \frac{-2 - 4}{5 - (-3)} = \frac{-6}{8} = -\frac{3}{4}

.

The slope of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by the vertical change divided by horizontal change.

2

Determine the slope of line

M

using the perpendicular line slope relationship.

m_M = -\frac{1}{m_L} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

3

Find the equation of line

M

using point-slope form

y - y_1 = m(x - x_1)

with point

(1, -1)

and slope

m_M = \frac{4}{3}

.

y - (-1) = \frac{4}{3}(x - 1) \implies y + 1 = \frac{4}{3}x - \frac{4}{3} \implies y = \frac{4}{3}x - \frac{7}{3}

.

The point-slope form allows writing the linear equation directly given one point and the slope.

4

Identify the

y

-intercept of line

M

.

The

y

-intercept is

(0, -\frac{7}{3})

, or simply

-\frac{7}{3}

.

In slope-intercept form

y = mx + b

, the constant term

b

represents the

y

-intercept.

Key Concept

Perpendicular lines have slopes that are negative reciprocals (

m_1 \cdot m_2 = -1

). The equation of a line can be determined using its slope and a given point.

Estimated Time:1m 30s

Question 287Question →

For all real numbers $a$ and $b$ , which of the following expressions are equivalent to $(a + b)^2 - (a - b)^2$ ? Select all that apply.

Select all that apply

4ab

(2a)(2b)

0

2a^2 + 2b^2

2ab

Show answer & explanation

Answer: $4ab$ ; $(2a)(2b)$

Answer

The expressions

4ab

and

(2a)(2b)

are equivalent to

(a + b)^2 - (a - b)^2

.

Expanding the squared expressions gives

(a+b)^2 = a^2 + 2ab + b^2

and

(a-b)^2 = a^2 - 2ab + b^2

. Subtracting the second expression from the first yields

(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab

. Since

(2a)(2b)

also equals

4ab

, both

4ab

and

(2a)(2b)

are mathematically equivalent to the original expression.

Step-by-Step Solution

1

Expand

(a + b)^2

using the binomial square formula.

(a + b)^2 = a^2 + 2ab + b^2

Squaring a sum produces a trinomial containing two squared terms and a middle product term.

2

Expand

(a - b)^2

using the binomial square formula.

(a - b)^2 = a^2 - 2ab + b^2

Squaring a difference produces a trinomial with a negative middle product term.

3

Subtract the second expanded trinomial from the first.

(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = a^2 - a^2 + 2ab + 2ab + b^2 - b^2 = 4ab

Distributing the negative sign across

(a^2 - 2ab + b^2)

changes the signs inside the second set of parentheses.

4

Compare the simplified result

4ab

against all given choices.

Both

4ab

and

(2a)(2b) = 4ab

are identical to the calculated result.

Algebraic multiplication shows

(2a)(2b) = 4ab

.

Key Concept

Binomial expansion and combination of like terms

Question 288Question →

If $x$ satisfies the equation $\frac{3x - 1}{2} - \frac{x + 4}{5} = \frac{x + 7}{10} + 2$ , which of the following statements about $x$ must be true? Indicate all such statements.

Select all that apply

x > 3

3x

is an integer

x

is a solution to the equation

3x - 10 = 0

x

is an integer

x < 3.2

Show answer & explanation

Answer: $x > 3$ ; $3x$ is an integer; $x$ is a solution to the equation $3x - 10 = 0$

Answer

The statements that

x > 3

, that

3x

is an integer, and that

x

is a solution to the equation

3x - 10 = 0

are all correct.

Solving the equation yields

x = \frac{10}{3} \approx 3.33

. The statement

x > 3

is correct since

3.33 > 3

. The statement that

3x

is an integer is correct because

3 \times \frac{10}{3} = 10

. The statement that

x

is a solution to

3x - 10 = 0

is correct because

3\left(\frac{10}{3}\right) - 10 = 0

.

Step-by-Step Solution

1

Clear denominators by multiplying the entire linear equation by the least common multiple, 10.

5(3x - 1) - 2(x + 4) = (x + 7) + 20

Eliminating fractions simplifies the expression into a standard linear form.

2

Distribute terms across parentheses.

15x - 5 - 2x - 8 = x + 7 + 20

Expand both sides while correctly applying sign rules.

3

Combine like terms on each side.

13x - 13 = x + 27

Group variable terms together and constant terms together.

4

Isolate the variable

x

.

12x = 40 \implies x = \frac{40}{12} = \frac{10}{3}

Subtract

x

and add

13

on both sides, then divide by

12

.

5

Evaluate each given statement using

x = \frac{10}{3}

.

The value

\frac{10}{3} \approx 3.33

is greater than

3

;

3\left(\frac{10}{3}\right) = 10

is an integer;

3\left(\frac{10}{3}\right) - 10 = 0

is true;

\frac{10}{3}

is not an integer;

\frac{10}{3}

is not less than

3.2

.

Determine which logical assertions hold true for

x = \frac{10}{3}

.

Key Concept

Solving multi-step linear equations with fractions and evaluating properties of rational solutions.

Question 289Question →

Line $Q$ passes through the coordinates $(-1, 7)$ and $(7, 1)$ in the Cartesian plane. Line $P$ is perpendicular to line $Q$ and intersects the $x$ -axis at $(-6, 0)$ . What is the $y$ -intercept of line $P$ ?

Show answer & explanation

Answer: 8

Answer

The

y

-intercept of line

P

is 8.

The slope of line

Q

is calculated as

\frac{1 - 7}{7 - (-1)} = -\frac{3}{4}

. Since line

P

is perpendicular to line

Q

, its slope is the negative reciprocal,

\frac{4}{3}

. Using the

x

-intercept point

(-6, 0)

, the line equation is

y - 0 = \frac{4}{3}(x + 6)

, which simplifies to

y = \frac{4}{3}x + 8

. Setting

x = 0

yields the

y

-intercept of

8

.

Step-by-Step Solution

1

Calculate the slope of line

Q

Slope of line

Q

is

-\frac{3}{4}

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{7 - (-1)} = \frac{-6}{8} = -\frac{3}{4}

.

2

Determine the slope of perpendicular line

P

Slope of line

P

is

\frac{4}{3}

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

m_P = -\frac{1}{m_Q} = -\frac{1}{-3/4} = \frac{4}{3}

.

3

Find the equation and

y

-intercept of line

P

The

y

-intercept is

8

Line

P

passes through

(-6, 0)

. Using point-slope form:

y - 0 = \frac{4}{3}(x - (-6)) \implies y = \frac{4}{3}x + 8

. Substituting

x = 0

gives

y = 8

.

Key Concept

Perpendicular lines and line equations in coordinate geometry

Question 290Question →

A delivery truck traveled from Warehouse X to Warehouse Y at a constant speed of $50$ miles per hour. On the return trip along the exact same route, heavy traffic reduced the truck's constant speed to $30$ miles per hour. If the total driving time for the entire round trip was $8$ hours, what was the distance, in miles, between Warehouse X and Warehouse Y?

120

150

160

250

300

Show answer & explanation

Answer: $150$

Answer

The distance between Warehouse X and Warehouse Y is

150

miles.

The correct distance between the two warehouses is

150

miles. Since time equals distance divided by rate, the time taken going to Warehouse Y is

\frac{d}{50}

hours and returning is

\frac{d}{30}

hours. Summing these expressions to equal the total

8

hours gives

\frac{d}{50} + \frac{d}{30} = 8

. Solving for

d

yields

\frac{8d}{150} = 8

, which simplifies to

d = 150

miles.

Step-by-Step Solution

1

Define variables and write time expressions for each leg of the trip

Let

d

be the distance in miles between Warehouse X and Warehouse Y. Time outbound is

t_1 = \frac{d}{50}

hours, and time inbound is

t_2 = \frac{d}{30}

hours.

Distance divided by rate gives the time taken for each individual leg of the journey.

2

Set up the equation using total elapsed time

\frac{d}{50} + \frac{d}{30} = 8

The total driving time for both legs combined is given as

8

hours.

3

Solve the algebraic equation for

d

\frac{3d + 5d}{150} = 8 \implies \frac{8d}{150} = 8 \implies 8d = 1200 \implies d = 150

Finding a common denominator of

150

allows combining the fractions to solve for the unknown distance

d

.

Key Concept

Distance-Rate-Time Relationship & Harmonic Mean Rate Modeling

Question 291Question →

If $x$ satisfies the linear equation $\frac{3(x - 1)}{2} - \frac{x + 5}{4} = \frac{2x - 3}{3}$ , what is the value of $4x + 7$ ?

15

19

23

27

31

Show answer & explanation

Answer: 19

Answer

19

Multiplying the entire equation by the least common denominator 12 yields

18(x - 1) - 3(x + 5) = 4(2x - 3)

. Expanding the terms gives

18x - 18 - 3x - 15 = 8x - 12

, which simplifies to

15x - 33 = 8x - 12

. Subtracting

8x

and adding

33

gives

7x = 21

, so

x = 3

. Substituting

x = 3

into

4x + 7

produces

4(3) + 7 = 19

.

Step-by-Step Solution

1

Find the least common denominator (LCD) for all fractions in the equation.

The LCD of 2, 4, and 3 is 12.

Clearing denominators simplifies the linear equation to integer coefficients.

2

Multiply both sides of the equation by 12.

12 \cdot \left(\frac{3(x - 1)}{2}\right) - 12 \cdot \left(\frac{x + 5}{4}\right) = 12 \cdot \left(\frac{2x - 3}{3}\right) \implies 6 \cdot 3(x - 1) - 3(x + 5) = 4(2x - 3)

Multiplying each term by 12 eliminates all fractions.

3

Expand the terms on both sides of the equation.

18(x - 1) - 3(x + 5) = 8x - 12 \implies 18x - 18 - 3x - 15 = 8x - 12

Distribute the coefficients across each set of parentheses.

4

Combine like terms and solve for

x

.

15x - 33 = 8x - 12 \implies 7x = 21 \implies x = 3

Isolate the variable term

x

on one side of the equation.

5

Evaluate the requested expression

4x + 7

using

x = 3

.

4(3) + 7 = 12 + 7 = 19

Substitute the value of

x

into

4x + 7

to find the final requested value.

Key Concept

Solving multi-step linear equations in one variable with fractional coefficients by clearing denominators.

Estimated Time:1m 30s

Question 292Question →

When the expression $\frac{3x^2 - 12}{x - 2}$ is completely simplified, it can be written in the form $ax + b$ for all $x \neq 2$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 9

Answer

The value of

a + b

is 9.

Factoring the numerator gives

3x^2 - 12 = 3(x^2 - 4) = 3(x - 2)(x + 2)

. Canceling the common factor

(x - 2)

in the denominator yields

3(x + 2)

, which expands to

3x + 6

. Matching this expression to

ax + b

reveals

a = 3

and

b = 6

. Adding these constants gives

a + b = 9

.

Step-by-Step Solution

1

Factor the numerator of the rational expression.

3x^2 - 12 = 3(x^2 - 4) = 3(x - 2)(x + 2)

Factoring out the common numerical factor 3 and using the difference of squares identity

(u^2 - v^2 = (u-v)(u+v))

completely factors the numerator.

2

Simplify the expression by canceling common factors.

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

Because

x \neq 2

, the factor

(x - 2)

is non-zero and can be canceled from both numerator and denominator.

3

Identify the values of

a

and

b

and compute

a + b

.

a = 3

,

b = 6

, so

a + b = 3 + 6 = 9

Comparing

3x + 6

to

ax + b

gives

a = 3

and

b = 6

.

Key Concept

Factoring algebraic expressions using common monomial factors and the difference of squares to simplify rational expressions.

Question 293Question →

Line $L$ is defined by the equation $3x - 4y = 12$ . Line $M$ is perpendicular to line $L$ and passes through the point $(6, -1)$ . Which of the following statements about line $M$ must be true? Select all that apply.

Select all that apply

Line

M

has a

y

-intercept of

(0, 7)

.

Line

M

passes through the point

(3, 3)

.

Line

M

passes through Quadrant III.

The

x

-intercept of line

M

is

(7, 0)

.

Line

M

is parallel to the line given by

4x + 3y = 10

.

Show answer & explanation

Answer: Line $M$ has a $y$ -intercept of $(0, 7)$ .; Line $M$ passes through the point $(3, 3)$ .; Line $M$ is parallel to the line given by $4x + 3y = 10$ .

Answer

The correct statements are that line

M

has a

y

-intercept of

(0, 7)

, passes through the point

(3, 3)

, and is parallel to the line given by

4x + 3y = 10

.

Line

L

has equation

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

, so its slope is

\frac{3}{4}

. The perpendicular line

M

has slope

-\frac{4}{3}

. Using point

(6, -1)

, the equation of line

M

is

y = -\frac{4}{3}x + 7

. The statement claiming a

y

-intercept of

(0, 7)

is correct because

y = 7

when

x = 0

. The statement claiming line

M

passes through

(3, 3)

is correct because

-\frac{4}{3}(3) + 7 = 3

. The statement claiming line

M

is parallel to

4x + 3y = 10

is correct because

4x + 3y = 10

has a slope of

-\frac{4}{3}

, which matches the slope of line

M

.

Step-by-Step Solution

1

Find the slope of line

L

.

Convert

3x - 4y = 12

to slope-intercept form:

4y = 3x - 12 \implies y = \frac{3}{4}x - 3

. The slope of line

L

is

m_L = \frac{3}{4}

.

The slope of a linear equation in standard form can be determined by expressing it as

y = mx + b

.

2

Determine the slope and equation of line

M

.

Since line

M

is perpendicular to line

L

, its slope is the negative reciprocal:

m_M = -\frac{4}{3}

. Using point-slope form with

(6, -1)

:

y - (-1) = -\frac{4}{3}(x - 6) \implies y + 1 = -\frac{4}{3}x + 8 \implies y = -\frac{4}{3}x + 7

.

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

3

Evaluate statement regarding the

y

-intercept.

Setting

x = 0

gives

y = 7

, so the

y

-intercept is

(0, 7)

. This statement is true.

The

y

-intercept occurs where

x = 0

.

4

Evaluate statement regarding point

(3, 3)

.

Plug in

x = 3

:

y = -\frac{4}{3}(3) + 7 = 3

. The point

(3, 3)

lies on line

M

. This statement is true.

A point lies on a line if its coordinates satisfy the line equation.

5

Evaluate statement regarding Quadrant III passage.

For

x < 0

,

y = -\frac{4}{3}x + 7 > 7 > 0

(Quadrant II). For

0 \le x \le 5.25

,

y \ge 0

(Quadrant I). For

x > 5.25

,

y < 0

(Quadrant IV). The line does not enter Quadrant III. This statement is false.

Lines with negative slopes and positive

y

-intercepts pass through Quadrants I, II, and IV only.

6

Evaluate statement regarding the

x

-intercept.

Setting

y = 0

:

0 = -\frac{4}{3}x + 7 \implies x = \frac{21}{4} = 5.25

, giving

x

-intercept

(\frac{21}{4}, 0)

. This statement is false.

The

x

-intercept occurs where

y = 0

.

7

Evaluate statement regarding parallelism to

4x + 3y = 10

.

4x + 3y = 10 \implies y = -\frac{4}{3}x + \frac{10}{3}

. The slope is

-\frac{4}{3}

, matching line

M

's slope, with distinct

y

-intercepts. This statement is true.

Lines with identical slopes and different

y

-intercepts are parallel.

Key Concept

Perpendicular and parallel line slope relationships, point-slope equation derivation, and coordinate plane quadrant navigation.

Question 294Question →

If $x$ is a positive real number such that $2x^2 - 5x - 3 = 0$ , what is the value of $x^2 + 1$ ?

4

\frac{5}{4}

9

10

26

Show answer & explanation

Answer: 10

Answer

10

Factoring the quadratic equation

2x^2 - 5x - 3 = 0

yields

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

, giving solutions

x = -\frac{1}{2}

and

x = 3

. Because

x

is specified as a positive real number, we choose

x = 3

. Substituting

x = 3

into

x^2 + 1

yields

3^2 + 1 = 10

.

Step-by-Step Solution

1

Factor the quadratic equation

2x^2 - 5x - 3 = 0

.

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

Find two linear factors whose product gives

2x^2 - 5x - 3

.

2

Solve for the roots of the equation.

x = -\frac{1}{2}

or

x = 3

Set each factor equal to zero:

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

and

x - 3 = 0 \implies x = 3

.

3

Apply the constraint

x > 0

to identify the valid root.

x = 3

The root

x = -\frac{1}{2}

is negative and thus violates the given condition that

x

is positive.

4

Substitute

x = 3

into the target expression

x^2 + 1

.

3^2 + 1 = 9 + 1 = 10

Evaluate the expression using the valid positive value of

x

.

Key Concept

Solving quadratic equations by factoring and applying domain constraints

Estimated Time:45s

Question 295Question →

In the $xy$ -plane, line $k$ passes through the point $(2, -3)$ and has a slope of $-\frac{4}{3}$ . Line $m$ is parallel to line $k$ and has an $x$ -intercept at $(6, 0)$ . What is the $y$ -intercept of line $m$ ?

8

-8

\frac{9}{2}

\frac{3}{4}

-6

Show answer & explanation

Answer: $8$

Answer

The

y

-intercept of line

m

is

8

.

Parallel lines have equal slopes, so line

m

has slope

m = -\frac{4}{3}

. An

x

-intercept of

(6, 0)

means line

m

passes through

(6, 0)

. Substituting

x = 6

,

y = 0

, and

m = -\frac{4}{3}

into

y = mx + b

gives

0 = -\frac{4}{3}(6) + b

, which simplifies to

0 = -8 + b

, so

b = 8

. Thus, the

y

-intercept is

8

.

Step-by-Step Solution

1

Determine the slope of line

m

The slope of line

m

is

-\frac{4}{3}

.

Parallel lines in the coordinate plane have identical slopes. Since line

k

has a slope of

-\frac{4}{3}

, line

m

must also have a slope of

-\frac{4}{3}

.

2

Use the

x

-intercept

(6, 0)

to solve for the

y

-intercept

b

of line

m

0 = -\frac{4}{3}(6) + b \implies 0 = -8 + b \implies b = 8

.

Substitute

x = 6

,

y = 0

, and slope

m = -\frac{4}{3}

into the slope-intercept form

y = mx + b

.

Key Concept

Parallel line slopes and slope-intercept form

Estimated Time:1m 15s

Question 296Question →

A specialty coffee shop prepares two custom coffee bean blends using Arabica and Robusta beans. The first 10-pound blend consists of 4 pounds of Arabica beans and 6 pounds of Robusta beans and costs $\$52$ . The second 10-pound blend consists of 7 pounds of Arabica beans and 3 pounds of Robusta beans and costs $\$61$ . What is the cost, in dollars, of 1 pound of Arabica coffee beans?

Show answer & explanation

Answer: 7

Answer

The cost of 1 pound of Arabica coffee beans is 7 dollars.

By setting up the system

4A + 6R = 52

and

7A + 3R = 61

, we can simplify the first equation to

2A + 3R = 26

. Subtracting this simplified equation from the second equation eliminates

R

, giving

5A = 35

, which simplifies directly to

A = 7

.

Step-by-Step Solution

1

Define variables and set up the system of linear equations.

Let

A

be the price per pound of Arabica beans and

R

be the price per pound of Robusta beans.
Equation 1:

4A + 6R = 52

Equation 2:

7A + 3R = 61

Translating the quantitative relationship given in the problem statement into algebraic equations.

2

Simplify Equation 1 and eliminate variable

R

by subtraction.

Dividing Equation 1 by 2 gives

2A + 3R = 26

. Subtracting this from Equation 2 yields

(7A + 3R) - (2A + 3R) = 61 - 26

, which reduces to

5A = 35

.

Matching coefficients of

R

allows for straightforward elimination of

R

.

3

Solve for variable

A

.

A = 7

Dividing 35 by 5 yields the price per pound of Arabica beans.

Key Concept

Solving a system of 2x2 linear equations using substitution or elimination.

Question 297Question →

A craft brewery blends two batches of cider. Batch A contains a $12\%$ sugar solution by volume, and Batch B contains a $20\%$ sugar solution by volume. The brewer mixes $15$ liters of Batch A with $x$ liters of Batch B to create a resulting mixture that is $17\%$ sugar by volume. What is the value of $x$ ?

Show answer & explanation

Answer: 25

Answer

The value of

x

is

25

.

Equating the total amount of pure sugar before and after mixing gives

0.12(15) + 0.20x = 0.17(15 + x)

. Simplifying yields

1.8 + 0.20x = 2.55 + 0.17x

, which reduces to

0.03x = 0.75

, giving

x = 25

.

Step-by-Step Solution

1

Calculate the volume of pure sugar contributed by Batch A

0.12 \times 15 = 1.8

liters of pure sugar

Batch A is

12\%

sugar by volume and has a total volume of

15

liters.

2

Express the total volume of pure sugar in terms of

x

Total sugar volume =

1.8 + 0.20x

liters

Batch B adds

x

liters of a

20\%

sugar solution.

3

Set up the mixture equation

1.8 + 0.20x = 0.17(15 + x)

The final mixture has a total volume of

(15 + x)

liters with a concentration of

17\%

sugar.

4

Expand and solve the linear equation for

x

1.8 + 0.20x = 2.55 + 0.17x \implies 0.03x = 0.75 \implies x = 25

Subtract

0.17x

and

1.8

from both sides, then divide by

0.03

.

Key Concept

Linear Equations in One Variable (Mixture Word Problems)

Estimated Time:1m 30s

Question 298Question →

For all real numbers $x$ such that $x \neq 5$ , which of the following expressions is equivalent to $\frac{x^2 - 25}{3x - 15}$ ?

\frac{x - 5}{3}

x + 5

\frac{x + 5}{3}

\frac{x - 25}{3}

\frac{x + 5}{15}

Show answer & explanation

Answer: $\frac{x + 5}{3}$

Answer

The expression

\frac{x + 5}{3}

is equivalent to the given rational expression for all

x \neq 5

.

Factoring the numerator as a difference of squares gives

(x - 5)(x + 5)

, and factoring the denominator gives

3(x - 5)

. Dividing out the common factor

(x - 5)

simplifies the expression to

\frac{x + 5}{3}

.

Step-by-Step Solution

1

Factor the numerator using the difference of squares identity.

x^2 - 25 = (x - 5)(x + 5)

The difference of squares

a^2 - b^2

factors into

(a - b)(a + b)

.

2

Factor out the greatest common factor from the denominator.

3x - 15 = 3(x - 5)

Both terms in

3x - 15

share a common factor of

3

.

3

Divide out the common binomial factor

(x - 5)

from the numerator and denominator.

\frac{(x - 5)(x + 5)}{3(x - 5)} = \frac{x + 5}{3}

Since

x \neq 5

, the factor

(x - 5)

is non-zero and can be cancelled.

Key Concept

Simplifying rational algebraic expressions by factoring difference of squares and common linear factors.

Question 299Question →

In the $xy$ -plane, line $k$ passes through the points $(-6, 0)$ and $(0, 4)$ . Line $m$ is perpendicular to line $k$ and intersects line $k$ at its $y$ -intercept. What is the $x$ -intercept of line $m$ ?

-\frac{8}{3}

-6

\frac{3}{2}

\frac{8}{3}

6

Show answer & explanation

Answer: $\frac{8}{3}$

Answer

The

x

-intercept of line

m

is

\frac{8}{3}

.

Line

k

has a slope of

\frac{4 - 0}{0 - (-6)} = \frac{2}{3}

. Because line

m

is perpendicular to line

k

, its slope is the negative reciprocal,

-\frac{3}{2}

. Line

m

intersects line

k

at its

y

-intercept

(0, 4)

, giving line

m

the equation

y = -\frac{3}{2}x + 4

. Setting

y = 0

yields

0 = -\frac{3}{2}x + 4

, which solves to

x = \frac{8}{3}

.

Step-by-Step Solution

1

Calculate the slope of line

k

.

The slope

m_k = \frac{4 - 0}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is calculated using

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

.

2

Determine the slope and

y

-intercept of line

m

.

The slope of line

m

is

m_m = -\frac{3}{2}

, and its

y

-intercept is

(0, 4)

.

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

m

intersects line

k

at

(0, 4)

.

3

Write the equation of line

m

and solve for its

x

-intercept.

Setting

y = 0

in

y = -\frac{3}{2}x + 4

yields

0 = -\frac{3}{2}x + 4 \implies \frac{3}{2}x = 4 \implies x = \frac{8}{3}

.

The

x

-intercept of a line is the value of

x

when

y = 0

.

Key Concept

Perpendicular line slopes and intercept calculations

Question 300Question →

A non-profit foundation distributes a total grant of $\$108,000$ among three research teams: Team A, Team B, and Team C. Team B receives $\$4,000$ more than Team A. Team C receives twice as much as the combined amount received by Team A and Team B. What amount does Team B receive?

\$16,000

\$20,000

\$24,000

\$28,000

\$72,000

Show answer & explanation

Answer: $\$20,000$

Answer

Team B receives

\$20,000

.

Defining Team A's share as

x

gives Team B a share of

x + 4,000

and Team C a share of

2(x + x + 4,000) = 4x + 8,000

. Combining these yields

6x + 12,000 = 108,000

, which solves to

x = 16,000

. Therefore, Team B receives

16,000 + 4,000 = \$20,000

.

Step-by-Step Solution

1

Define variables for each team's share in terms of a single unknown variable.

Let

x

be the amount received by Team A in dollars. Then Team B receives

x + 4,000

, and Team C receives

2 \cdot [x + (x + 4,000)] = 2(2x + 4,000) = 4x + 8,000

.

Expressing all quantities in terms of

x

reduces the problem to a linear equation in one variable.

2

Set up the linear equation representing the total grant amount.

x + (x + 4,000) + (4x + 8,000) = 108,000 \implies 6x + 12,000 = 108,000

.

The sum of the individual shares must equal the total grant of

\$108,000

.

3

Solve the linear equation for

x

.

6x = 96,000 \implies x = 16,000

.

Subtract

12,000

from both sides and divide by

6

.

4

Calculate Team B's share using the value of

x

.

Team B's share

= x + 4,000 = 16,000 + 4,000 = 20,000

.

The question specifically asks for Team B's share, not Team A's.

Key Concept

Formulating and solving a linear equation in one variable from a multi-step algebraic word problem.

Estimated Time:1m 30s

Algebra

Answer

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