Algebra Practice Questions — GRE General Test — Page 16

Question 301Question →

If $x$ satisfies the equation $\frac{5(x - 2)}{3} - \frac{2x - 1}{4} = \frac{x + 7}{6}$ , which of the following statements must be true? Select all that apply.

Select all that apply

x > 4

4x

is an integer

x

is an integer

2x + 1 > 10

3x - 2 = 10.75

Show answer & explanation

Answer: $x > 4$ ; $4x$ is an integer; $3x - 2 = 10.75$

Answer

The correct statements are that

x > 4

,

4x

is an integer, and

3x - 2 = 10.75

.

Solving the linear equation yields

x = 4.25

or

\frac{17}{4}

. Using this value:

4.25 > 4

is true;

4 \times \frac{17}{4} = 17

is an integer, so that statement is true; and

3(4.25) - 2 = 10.75

is also true.

Step-by-Step Solution

1

Find a common denominator to clear fractions in the given equation.

The least common multiple of denominators

3

,

4

, and

6

is

12

.

Clearing denominators simplifies multi-term fractional equations into standard linear form.

2

Multiply every term of the equation by

12

and expand numerators.

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

.

Multiplying each term by

12

eliminates all fraction bars.

3

Distribute the constants through the parentheses on both sides.

20x - 40 - 6x + 3 = 2x + 14

.

Applying the distributive property correctly accounts for negative signs across parentheses.

4

Combine like terms on the left side and solve for

x

.

14x - 37 = 2x + 14 \implies 12x = 51 \implies x = \frac{51}{12} = \frac{17}{4} = 4.25

.

Isolating

x

gives the exact rational value of the solution.

5

Evaluate each provided statement using

x = 4.25

.

The statement

x > 4

is true (

4.25 > 4

). The statement

4x

is an integer is true (

4 \times 4.25 = 17

). The statement

3x - 2 = 10.75

is true (

3 \times 4.25 - 2 = 10.75

). The other two statements are false.

Direct substitution verifies which conditions hold.

Key Concept

Solving multi-step linear equations in one variable by clearing denominators and combining variable terms.

Estimated Time:1m 30s

Question 302Question →

In the $xy$ -plane, line $L$ passes through the origin $(0, 0)$ and the point $P(a, b)$ , where $a > 0$ and $b > 0$ . Line $M$ is perpendicular to line $L$ at point $P$ . If line $M$ has a $y$ -intercept at $(0, 10)$ and $b = 2$ , what is the value of $a$ ?

2\sqrt{2}

4

4\sqrt{2}

8

16

Show answer & explanation

Answer: $4$

Answer

The value of

a

is

4

.

Line

L

connects

(0,0)

to

(a,2)

, giving it a slope of

\frac{2}{a}

. Because line

M

is perpendicular to line

L

, its slope must be the negative reciprocal,

-\frac{a}{2}

. Line

M

also connects

(a,2)

to its

y

-intercept

(0,10)

, so its slope can independently be written as

\frac{10-2}{0-a} = -\frac{8}{a}

. Setting

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

yields

a^2 = 16

. Since

a > 0

,

a = 4

.

Step-by-Step Solution

1

Find the slope of line

L

Since line

L

passes through

(0, 0)

and

P(a, 2)

, its slope is

m_L = \frac{2 - 0}{a - 0} = \frac{2}{a}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Determine the slope of perpendicular line

M

The slope of line

M

is

m_M = -\frac{1}{m_L} = -\frac{a}{2}

.

Perpendicular lines have negative reciprocal slopes.

3

Calculate the slope of line

M

using given points

(a, 2)

and

(0, 10)

Using the slope formula:

m_M = \frac{10 - 2}{0 - a} = \frac{8}{-a} = -\frac{8}{a}

.

Line

M

passes through the point of intersection

P(a, 2)

and its

y

-intercept

(0, 10)

.

4

Equate the two slope expressions and solve for

a

-\frac{a}{2} = -\frac{8}{a} \implies a^2 = 16 \implies a = 4

(since

a > 0$).

Both expressions represent the slope of line

M

.

Key Concept

Perpendicular Slopes and Line Equations in Coordinate Geometry

Estimated Time:1m 30s

Question 303Question →

In the $xy$ -plane, line $k$ passes through the points $(-3, 5)$ and $(3, 1)$ . Line $m$ is perpendicular to line $k$ and passes through the point $(1, -1)$ . Which of the following statements must be true? Indicate all such statements.

Select all that apply

The slope of line

m

is

\frac{3}{2}

.

Line

m

passes through Quadrants I, III, and IV.

The

x

-intercept of line

k

is

\left(\frac{9}{2}, 0\right)

.

The slope of line

k

is

-\frac{3}{2}

.

Line

k

and line

m

intersect at the point

(3, 1)

.

Show answer & explanation

Answer: The slope of line $m$ is $\frac{3}{2}$ .; Line $m$ passes through Quadrants I, III, and IV.; The $x$ -intercept of line $k$ is $\left(\frac{9}{2}, 0\right)$ .

Answer

The statements confirming that the slope of line

m

is

\frac{3}{2}

, line

m

passes through Quadrants I, III, and IV, and the

x

-intercept of line

k

is

\left(\frac{9}{2}, 0\right)

are all correct.

The slope of line

k

is

-\frac{2}{3}

, making line

m

's perpendicular slope

\frac{3}{2}

. The equation for line

m

is

y = \frac{3}{2}x - \frac{5}{2}

, which crosses the

y

-axis at

\left(0, -\frac{5}{2}\right)

and the

x

-axis at

\left(\frac{5}{3}, 0\right)

, thus entering Quadrants I, III, and IV. The

x

-intercept of line

k

is found by setting

y = 0

in

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

, yielding

x = \frac{9}{2}

.

Step-by-Step Solution

1

Calculate the slope of line

k

using the slope formula

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

.

Slope of line

k

is

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

.

Determining the slope of line

k

is required to find perpendicular slopes and line equations.

2

Determine the slope and equation of line

m

.

Perpendicular slope is

m_m = -\frac{1}{m_k} = \frac{3}{2}

. Using point-slope form with point

(1, -1)

,

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

.

Perpendicular lines have negative reciprocal slopes.

3

Find the

x

-intercept of line

k

and analyze the quadrant passage of line

m

.

Line

k

equation is

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

; setting

y = 0

yields

x = \frac{9}{2}

. For line

m

, positive slope and negative

y

-intercept

\left(0, -\frac{5}{2}\right)

mean it traverses Quadrants I, III, and IV.

Setting

y=0

gives the

x

-intercept, and evaluating the line equation across negative, zero, and positive

x

-values identifies quadrant coverage.

Key Concept

Coordinate Geometry: Line equations, slopes of perpendicular lines, intercepts, and quadrant passage.

Question 304Question →

A technical consultant charges a one-time setup fee of $\$150$ plus a standard rate of $\$45$ per hour for the first $20$ hours of work on a project. For any hours worked beyond $20$ hours, the consultant charges an increased hourly rate that is $20\%$ higher than the standard rate. If the total bill for a project was $\$1,320$ , how many total hours did the consultant work on the project?

Show answer & explanation

Answer: 25

Answer

The consultant worked a total of 25 hours on the project.

To find the total hours worked, first subtract the setup fee (

150) and the cost of the first 20 hours (20 *

45 =

900) from the total bill of

1,320. This leaves

1,320 -

1,050 =

270. The hourly rate after 20 hours increases by 20% to

45 * 1.20 =

54 per hour. Dividing the remaining

270 by $54 gives 5 additional hours. Adding these 5 hours to the initial 20 hours gives a total of 25 hours.

Step-by-Step Solution

1

Determine the base cost for the setup fee and the initial 20 hours.

Base cost =

150 + (20 *

45) = $1,050.

The initial cost tier applies up to 20 hours of work.

2

Calculate the higher hourly rate applied to additional hours worked beyond 20.

Increased rate =

45 * 1.20 =

54 per hour.

The rate increases by 20% over the standard rate of $45 per hour.

3

Formulate and solve a linear equation for total hours

h

.

1050 + 54(h - 20) = 1320 \implies 54(h - 20) = 270 \implies h - 20 = 5 \implies h = 25

.

Subtracting the base cost leaves

270 for overtime hours, which divides by

54 per hour to yield 5 additional hours, for 25 hours total.

Key Concept

Linear Equations in One Variable

Question 305Question →

Which of the following values are solutions to the equation $(x - 3)^2 = 16$ ? Select all that apply.

Select all that apply

-7

-1

5

7

19

Show answer & explanation

Answer: $-1$ ; $7$

Answer

The solutions to the equation are

-1

and

7

.

Taking the square root of both sides of

(x - 3)^2 = 16

gives

x - 3 = 4

or

x - 3 = -4

. Solving these equations yields

x = 7

and

x = -1

.

Step-by-Step Solution

1

Take the square root of both sides of the quadratic equation

(x - 3)^2 = 16

.

x - 3 = \pm \sqrt{16} = \pm 4

Applying the square root property yields both a positive and a negative root.

2

Solve the equation corresponding to the positive root:

x - 3 = 4

.

x = 4 + 3 = 7

Adding

3

to both sides isolates

x

.

3

Solve the equation corresponding to the negative root:

x - 3 = -4

.

x = -4 + 3 = -1

Adding

3

to both sides isolates

x

for the negative root.

Key Concept

Solving quadratic equations of the form

(x - a)^2 = k

using the square root property.

Estimated Time:1m 0s

Question 306Question →

A non-profit organization hosted two fundraising events, Event A and Event B. Event A charged a ticket price of $\$30$ per person and collected an additional fixed donation of $\$250$ from a local sponsor. Event B charged a ticket price of $\$45$ per person and collected a fixed donation of $\$400$ from a corporate sponsor. The number of attendees at Event B was $10$ fewer than twice the number of attendees at Event A. If the total revenue raised from both events combined was $\$8,600$ , how many people attended Event B?

60

70

130

140

150

Show answer & explanation

Answer: 130

Answer

130 people attended Event B.

Setting up the linear equation for total revenue gives

(30x + 250) + (45(2x - 10) + 400) = 8,600

, where

x

is the attendance at Event A. Expanding and combining like terms yields

120x + 200 = 8,600

, which solves to

x = 70

. Substituting

x = 70

into the expression for Event B attendance (

2x - 10

) gives

2(70) - 10 = 130

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

x

be the number of attendees at Event A. The number of attendees at Event B is

2x - 10

.

Event B has 10 fewer attendees than twice Event A.

2

Express the revenue generated by each event in terms of

x

.

Event A revenue:

30x + 250

; Event B revenue:

45(2x - 10) + 400 = 90x - 450 + 400 = 90x - 50

.

Revenue equals ticket price times attendees plus fixed sponsor donations.

3

Set up and simplify the linear equation for total combined revenue.

(30x + 250) + (90x - 50) = 8,600 \implies 120x + 200 = 8,600

.

The sum of revenues from both events equals the total revenue of $8,600.

4

Solve the linear equation for

x

.

120x = 8,400 \implies x = 70

.

Subtract 200 from both sides and divide by 120 to isolate

x

.

5

Calculate the number of attendees at Event B.

Event B attendees

= 2(70) - 10 = 140 - 10 = 130

.

Substitute

x = 70

into the expression

2x - 10

.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 307Question →

For all real numbers

x

such that

x \neq 3

and

x \neq -3

, the algebraic expression

\frac{\frac{x^4 - 81}{x^2 - 9} \cdot \frac{x^3 - 27}{x^2 + 3x + 9} - x(x - 3)^2}{x^2 - 9}

simplifies to a constant value. What is the value of this constant?

Show answer & explanation

Answer: 3

Answer

The simplified expression evaluates to the constant value

3

.

Factoring the numerator components using difference of squares and difference of cubes simplifies the product term to

x^3 - 3x^2 + 9x - 27

. Subtracting

x(x - 3)^2 = x^3 - 6x^2 + 9x

simplifies the entire numerator to

3x^2 - 27 = 3(x^2 - 9)

. Dividing by the denominator

(x^2 - 9)

cancels out the variable terms entirely, yielding the constant value 3.

Step-by-Step Solution

1

Simplify the first rational component using the difference of squares identity

\frac{x^4 - 81}{x^2 - 9} = \frac{(x^2 - 9)(x^2 + 9)}{x^2 - 9} = x^2 + 9

Since

x \neq \pm 3

,

x^2 - 9 \neq 0

, allowing direct cancellation of

(x^2 - 9)

.

2

Simplify the second rational component using the difference of cubes identity

\frac{x^3 - 27}{x^2 + 3x + 9} = \frac{(x - 3)(x^2 + 3x + 9)}{x^2 + 3x + 9} = x - 3

Applying

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

where

a = x

and

b = 3

allows cancellation of the quadratic factor.

3

Multiply the simplified expressions

(x^2 + 9)(x - 3) = x^3 - 3x^2 + 9x - 27

Distribute each term of the binomials to get the expanded polynomial.

4

Expand the subtracted term in the numerator

x(x - 3)^2 = x(x^2 - 6x + 9) = x^3 - 6x^2 + 9x

Expand

(x - 3)^2 = x^2 - 6x + 9

and distribute

x

.

5

Subtract the two expanded expressions to simplify the entire numerator

(x^3 - 3x^2 + 9x - 27) - (x^3 - 6x^2 + 9x) = 3x^2 - 27 = 3(x^2 - 9)

Combine like terms;

x^3

and

9x

terms cancel out, leaving

3x^2 - 27

.

6

Divide the simplified numerator by the main denominator

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

Cancel the common factor

(x^2 - 9)

from numerator and denominator.

Key Concept

Simplifying complex algebraic expressions via polynomial factoring (difference of squares and difference of cubes) and combining like terms.

Question 308Question →

In the $xy$ -plane, line $k$ passes through the point $(3, 5)$ and has a slope of $-3$ . Line $m$ is perpendicular to line $k$ and also passes through the point $(3, 5)$ . What is the $x$ -intercept of line $m$ ?

-12

4

18

\frac{14}{3}

\frac{4}{3}

Show answer & explanation

Answer: $-12$

Answer

-12

The slope of line

k

is

-3

, so the slope of perpendicular line

m

is its negative reciprocal,

\frac{1}{3}

. Substituting slope

\frac{1}{3}

and point

(3, 5)

into point-slope form gives

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

, which simplifies to

y = \frac{1}{3}x + 4

. Setting

y = 0

to find the

x

-intercept gives

0 = \frac{1}{3}x + 4

, yielding

x = -12

.

Step-by-Step Solution

1

Determine the slope of line

m

Since line

m

is perpendicular to line

k

, its slope is the negative reciprocal of

-3

, which is

\frac{1}{3}

.

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

2

Find the equation of line

m

Using point-slope form with point

(3, 5)

and slope

\frac{1}{3}

:

y - 5 = \frac{1}{3}(x - 3) \implies y = \frac{1}{3}x + 4

.

A line with slope

m

passing through

(x_1, y_1)

follows

y - y_1 = m(x - x_1)

.

3

Calculate the

x

-intercept of line

m

Set

y = 0

:

0 = \frac{1}{3}x + 4 \implies \frac{1}{3}x = -4 \implies x = -12

.

The

x

-intercept is the

x

-coordinate where the line intersects the

x

-axis (

y = 0

).

Key Concept

Perpendicular Slopes and Line Intercepts

Estimated Time:1m 30s

Question 309Question →

If $a$ and $b$ are distinct real numbers, what is the simplified form of the algebraic expression $\frac{a^3 - b^3 - a(a^2 - b^2) + b(a - b)^2}{a - b}$ ?

ab + 2b^2

-ab

ab

a^2 + b^2

a(a + b)

Show answer & explanation

Answer: $ab$

Answer

ab

Factoring the common binomial term

(a - b)

from each term in the numerator yields

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

. Expanding inside the brackets gives

a^2 + ab + b^2 - a^2 - ab + ab - b^2

, which reduces completely to

ab

. Dividing

(a - b)(ab)

by

(a - b)

leaves the simplified expression

ab

.

Step-by-Step Solution

1

Apply standard algebraic factoring identities to each component of the numerator.

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

,

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

, and

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

.

Rewriting each grouping with the common binomial factor

(a - b)

allows for structured factoring of the numerator.

2

Factor out

(a - b)

from the entire numerator.

Numerator

= (a - b) \left[ (a^2 + ab + b^2) - a(a + b) + b(a - b) \right]

.

Extracting the common factor isolates the remaining polynomial terms inside brackets.

3

Expand and collect like terms within the bracketed expression.

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

.

Canceling opposite terms (

a^2 - a^2 = 0

,

ab - ab = 0

,

b^2 - b^2 = 0

) leaves only

ab

inside the brackets.

4

Divide the factored numerator by the denominator

(a - b)

.

\frac{(a - b)(ab)}{a - b} = ab

.

Since

a

and

b

are distinct (

a \neq b

),

a - b \neq 0

, allowing non-zero cancellation.

Key Concept

Simplifying complex algebraic expressions by factoring common binomial terms and applying algebraic identities

Question 310Question →

A bakery produces sourdough loaves using a mixture of two types of flour, Flour X and Flour Y. A standard batch requires a total of $84$ kilograms of flour. The amount of Flour X used is $12$ kilograms more than twice the amount of Flour Y used. If $y$ represents the amount of Flour Y, in kilograms, used in one standard batch, which of the following statements must be true? Select all such statements.

Select all that apply

y

is a positive integer multiple of

8

.

The ratio of the amount of Flour X to the amount of Flour Y used in one batch is

5:2

.

Flour X accounts for

75\%

of the total flour used in one batch.

The amount of Flour X used in one batch exceeds the amount of Flour Y used by

36

kilograms.

The combined amount of Flour Y used in

4

standard batches is

100

kilograms.

Show answer & explanation

Answer: $y$ is a positive integer multiple of $8$ .; The ratio of the amount of Flour X to the amount of Flour Y used in one batch is $5:2$ .; The amount of Flour X used in one batch exceeds the amount of Flour Y used by $36$ kilograms.

Answer

The correct statements are that

y

is a positive integer multiple of

8

, the ratio of Flour X to Flour Y used in one batch is

5:2

, and the amount of Flour X used in one batch exceeds the amount of Flour Y used by

36

kilograms.

Solving

y + (2y + 12) = 84

yields

y = 24

kilograms of Flour Y and

60

kilograms of Flour X. Since

24

is divisible by

8

, the statement asserting

y

is a multiple of

8

is true. The ratio of Flour X to Flour Y is

60:24 = 5:2

, making the ratio statement true. Finally,

60 - 24 = 36

, confirming that Flour X exceeds Flour Y by

36

kilograms.

Step-by-Step Solution

1

Set up the linear equation in terms of

y

.

Amount of Flour Y =

y

, Amount of Flour X =

2y + 12

. Total amount =

y + (2y + 12) = 84

.

The problem states Flour X is

12

kg more than twice Flour Y, and their sum equals

84

kg.

2

Solve the linear equation for

y

.

3y + 12 = 84 \implies 3y = 72 \implies y = 24

.

Subtract

12

from both sides and divide by

3

to isolate the variable

y

.

3

Calculate the amount of Flour X used.

Amount of Flour X =

2(24) + 12 = 48 + 12 = 60

kilograms.

Substitute

y = 24

back into the expression for Flour X.

4

Evaluate each given statement.

1)

y = 24

, which is a multiple of

8

(

24 = 8 \times 3

). (True)
2) Ratio of Flour X to Flour Y is

60 : 24 = 5 : 2

. (True)
3) Percentage of Flour X is

\frac{60}{84} = \frac{5}{7} \approx 71.43\% \neq 75\%

. (False)
4) Difference is

60 - 24 = 36

kg. (True)
5)

4

batches of Flour Y equal

4 \times 24 = 96

kg

\neq 100

kg. (False)

Direct calculation confirms which statements match the derived values.

Key Concept

Formulating and solving a linear equation in one variable from a word problem scenario, followed by logical evaluation of derived quantities.

Question 311Question →

If $x > 0$ and $x^2 - 6x - 16 = 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 8

Answer

The correct value of

x

is 8.

Factoring

x^2 - 6x - 16 = 0

gives

(x - 8)(x + 2) = 0

. Setting each factor to zero yields

x = 8

and

x = -2

. Given the constraint

x > 0

, the only valid answer is 8.

Step-by-Step Solution

1

Factor the quadratic expression

(x - 8)(x + 2) = 0

Find two numbers that multiply to

-16

and add to

-6

, which are

-8

and

2

.

2

Solve for possible values of

x

x = 8

or

x = -2

By the zero-product property, if

(x - 8)(x + 2) = 0

, then either

x - 8 = 0

or

x + 2 = 0

.

3

Apply the given constraint

x > 0

x = 8

The solution

x = -2

is rejected because

x

must be positive.

Key Concept

Factoring Quadratic Equations

Estimated Time:45s

Question 312Question →

A manufacturing plant operates two assembly lines, Line X and Line Y. Line X produces 15 units per hour, and Line Y produces 22 units per hour. On a certain day, Line X operated for 3 hours longer than Line Y did, and the two lines produced a total of 415 units. How many hours did Line Y operate?

Show answer & explanation

Answer: 10

Answer

10

Let

h

represent the number of hours Line Y operated. Because Line X operated for 3 hours longer than Line Y, Line X operated for

h + 3

hours. The total number of units produced by both lines is the sum of their individual outputs:

15(h + 3) + 22h = 415

. Distributing 15 yields

15h + 45 + 22h = 415

. Combining like terms gives

37h + 45 = 415

. Subtracting 45 from both sides yields

37h = 370

, and dividing by 37 gives

h = 10

. Therefore, Line Y operated for 10 hours.

Step-by-Step Solution

1

Define the variable representing Line Y's operating time in hours.

Let

h

represent the number of hours Line Y operated. Line X's operating time is

h + 3

hours.

Line X operated for 3 hours longer than Line Y.

2

Formulate a linear equation in one variable for total units produced.

15(h + 3) + 22h = 415

Total production is the sum of production from Line X (

15 \times (h + 3)

) and Line Y (

22 \times h

).

3

Distribute and combine like terms on the left side of the equation.

15h + 45 + 22h = 415 \implies 37h + 45 = 415

Apply the distributive property and combine variable terms.

4

Isolate the variable

h

.

37h = 370 \implies h = 10

Subtract 45 from both sides of the equation and divide by 37.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 313Question →

Given non-zero real numbers $x$ and $y$ where $|x| \neq |y|$ , simplify the complex rational expression:

\frac{\frac{x^3 + y^3}{x^2 - y^2} - \frac{x^2y - xy^2}{(x - y)^2}}{\frac{x^4 - y^4}{x^3 + x^2y + xy^2 + y^3}}

Which of the following represents the completely simplified expression?

1

x - y

x + y

\frac{x - y}{x + y}

\frac{x^2 + y^2}{x - y}

Show answer & explanation

Answer: 1

Answer

1

Both the entire complex numerator and the entire complex denominator independently simplify to

x - y

. Consequently, dividing the numerator

x - y

by the denominator

x - y

gives

1

.

Step-by-Step Solution

1

Simplify the first term of the main numerator

\frac{x^3 + y^3}{x^2 - y^2} = \frac{(x + y)(x^2 - xy + y^2)}{(x - y)(x + y)} = \frac{x^2 - xy + y^2}{x - y}

Factor the sum of cubes in the numerator and the difference of squares in the denominator, then cancel the common factor

(x + y)

.

2

Simplify the second term of the main numerator

\frac{x^2y - xy^2}{(x - y)^2} = \frac{xy(x - y)}{(x - y)^2} = \frac{xy}{x - y}

Factor out the greatest common factor

xy

from the numerator and cancel one factor of

(x - y)

.

3

Subtract the simplified terms in the main numerator

\frac{x^2 - xy + y^2}{x - y} - \frac{xy}{x - y} = \frac{x^2 - 2xy + y^2}{x - y} = \frac{(x - y)^2}{x - y} = x - y

Combine the numerators over the common denominator

(x - y)

, factor the perfect square trinomial

x^2 - 2xy + y^2 = (x - y)^2

, and simplify.

4

Simplify the main denominator

\frac{x^4 - y^4}{x^3 + x^2y + xy^2 + y^3} = \frac{(x - y)(x + y)(x^2 + y^2)}{x^2(x + y) + y^2(x + y)} = \frac{(x - y)(x + y)(x^2 + y^2)}{(x + y)(x^2 + y^2)} = x - y

Factor the numerator using difference of squares twice, factor the denominator by grouping, and cancel common factors

(x + y)(x^2 + y^2)

.

5

Divide the main numerator by the main denominator

\frac{x - y}{x - y} = 1

Divide the simplified main numerator (

x - y

) by the simplified main denominator (

x - y

).

Key Concept

Multi-step algebraic expression simplification using special factoring identities (sum/difference of cubes, difference of squares, quadratic trinomials, and factoring by grouping).

Estimated Time:2m 0s

Question 314Question →

A digital archive charges an annual subscription fee of $\$120$ , which includes $50$ free document downloads per year. For each additional document downloaded beyond the first $50$ , the archive charges a flat rate of $\$1.50$ . If a research group paid a total of $\$231$ to the archive last year, how many total documents did the research group download?

24

74

114

124

204

Show answer & explanation

Answer: 124

Answer

The total number of documents downloaded by the research group last year was 124.

Subtracting the base subscription fee of

120 from the total payment of

231 leaves

111 spent strictly on extra downloads. Dividing

111 by the $1.50 per-document rate yields 74 extra downloads. Adding the 50 included free downloads gives a total of 124 documents downloaded.

Step-by-Step Solution

1

Define the linear equation representing total cost.

120 + 1.50(x - 50) = 231

, where

x

represents the total number of documents downloaded.

The base subscription fee is

120, and the per-document rate of

1.50 applies only to documents in excess of the initial 50 free documents.

2

Isolate the variable term by subtracting the base subscription fee from both sides.

1.50(x - 50) = 231 - 120 = 111

This determines the portion of the total cost that resulted strictly from additional downloads.

3

Solve for the number of additional documents

(x - 50)

.

x - 50 = \frac{111}{1.50} = 74

Dividing the extra cost of

111 by the per-document rate of

1.50 gives the exact count of extra downloads.

4

Add the initial 50 free documents to find the total document count

x

.

x = 74 + 50 = 124

The question asks for the total documents downloaded, which combines the 50 free downloads and the 74 additional paid downloads.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 315Question →

For all real numbers $x$ and $y$ such that $x^2 + xy + y^2 \neq 0$ , which of the following expressions are equivalent to $\frac{x^6 - y^6}{x^2 + xy + y^2}$ ? Select all such expressions.

Select all that apply

(x^2 - y^2)(x^2 - xy + y^2)

(x - y)(x^3 + y^3)

x^4 - x^3 y + x y^3 - y^4

(x + y)(x^3 - y^3)

x^4 - y^4

Show answer & explanation

Answer: $(x^2 - y^2)(x^2 - xy + y^2)$ ; $(x - y)(x^3 + y^3)$ ; $x^4 - x^3 y + x y^3 - y^4$

Answer

The equivalent expressions are

(x^2 - y^2)(x^2 - xy + y^2)

,

(x - y)(x^3 + y^3)

, and

x^4 - x^3y + xy^3 - y^4

.

Factoring the numerator

x^6 - y^6

as a difference of squares yields

(x^3 - y^3)(x^3 + y^3)

. Applying the difference of cubes identity

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

enables cancellation of the non-zero denominator

x^2 + xy + y^2

, leaving

(x - y)(x^3 + y^3)

. Expanding this product gives

x^4 - x^3y + xy^3 - y^4

. Furthermore, factoring

x^3 + y^3

as

(x + y)(x^2 - xy + y^2)

and regrouping

(x - y)(x + y)

gives

(x^2 - y^2)(x^2 - xy + y^2)

. Consequently, the three valid equivalent forms are

(x^2 - y^2)(x^2 - xy + y^2)

,

(x - y)(x^3 + y^3)

, and

x^4 - x^3y + xy^3 - y^4

.

Step-by-Step Solution

1

Factor the numerator

x^6 - y^6

as a difference of squares.

x^6 - y^6 = (x^3)^2 - (y^3)^2 = (x^3 - y^3)(x^3 + y^3)

Applying the difference of squares identity

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

with

a = x^3

and

b = y^3

.

2

Apply the difference of cubes identity to

x^3 - y^3

.

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

This reveals the non-zero quadratic factor present in the denominator.

3

Substitute into the original fraction and cancel the common factor

(x^2 + xy + y^2)

.

\frac{(x - y)(x^2 + xy + y^2)(x^3 + y^3)}{x^2 + xy + y^2} = (x - y)(x^3 + y^3)

Since

x^2 + xy + y^2 \neq 0

, dividing numerator and denominator by

(x^2 + xy + y^2)

simplifies the expression to

(x - y)(x^3 + y^3)

.

4

Expand

(x - y)(x^3 + y^3)

to check for equivalent expanded polynomial forms.

(x - y)(x^3 + y^3) = x^4 + xy^3 - x^3y - y^4 = x^4 - x^3y + xy^3 - y^4

Distributing terms verifies polynomial equivalence.

5

Factor

x^3 + y^3

and regroup to find another equivalent factored representation.

(x - y)(x^3 + y^3) = (x - y)(x + y)(x^2 - xy + y^2) = (x^2 - y^2)(x^2 - xy + y^2)

Applying the sum of cubes identity

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

and combining

(x - y)(x + y) = x^2 - y^2

.

Key Concept

Factoring higher-degree algebraic expressions using difference of squares and sum/difference of cubes identities

Question 316Question →

For all real numbers $x$ and $y$ such that $x \neq y$ and $x \neq -y$ , which of the following is equivalent to the algebraic expression $\frac{x^3(x + 2y) - y^3(y + 2x)}{x^2 - y^2}$ ?

x^2 + y^2

(x + y)^2

(x - y)^2

x^2 + 2xy - y^2

x + y

Show answer & explanation

Answer: $(x + y)^2$

Answer

The simplified expression is

(x + y)^2

.

Expanding and grouping the terms in the numerator gives

(x^4 - y^4) + 2xy(x^2 - y^2) = (x^2 - y^2)(x^2 + 2xy + y^2) = (x^2 - y^2)(x + y)^2

. Canceling the common factor

(x^2 - y^2)

from both the numerator and the denominator leaves

(x + y)^2

.

Step-by-Step Solution

1

Expand the numerator terms

x^3(x + 2y) - y^3(y + 2x) = x^4 + 2x^3y - y^4 - 2xy^3

Apply the distributive property to remove parentheses in the numerator.

2

Group the terms in the numerator to factor

(x^4 - y^4) + (2x^3y - 2xy^3) = (x^2 - y^2)(x^2 + y^2) + 2xy(x^2 - y^2)

Use difference of squares on

x^4 - y^4

and factor out the greatest common factor

2xy

from the remaining terms.

3

Factor out the common term

(x^2 - y^2)

from the numerator

(x^2 - y^2)(x^2 + y^2 + 2xy) = (x^2 - y^2)(x + y)^2

Recognize that

x^2 + 2xy + y^2

is the perfect square binomial

(x + y)^2

.

4

Simplify the rational expression by canceling common factors

\frac{(x^2 - y^2)(x + y)^2}{x^2 - y^2} = (x + y)^2

Divide numerator and denominator by

(x^2 - y^2)

, which is non-zero since

x \neq \pm y

.

Key Concept

Factoring high-degree algebraic expressions by grouping terms, recognizing difference of squares, and applying perfect square binomial identities.

Estimated Time:2m 0s

Question 317Question →

If $x$ is the solution to the linear equation $\frac{4x - 3}{3} - \frac{x + 2}{4} = \frac{5x}{6}$ , which of the following statements about $x$ must be true? Select all such statements.

Select all that apply

x

is a multiple of 3

2x - 5 > 6

x^2 - 5x = 6

x

is a prime number

x < 5

Show answer & explanation

Answer: $x$ is a multiple of 3; $2x - 5 > 6$ ; $x^2 - 5x = 6$

Answer

The statements establishing that

x

is a multiple of 3, that

2x - 5 > 6

, and that

x^2 - 5x = 6

are all true.

Solving the linear equation yields

x = 6

. Substituting

x = 6

into each statement shows that

x

is a multiple of 3 (

6 = 3 \times 2

),

2x - 5 > 6

evaluates to

7 > 6

, and

x^2 - 5x = 6

evaluates to

36 - 30 = 6

. All three statements are correct.

Step-by-Step Solution

1

Clear denominators by multiplying both sides by the least common multiple.

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

The least common multiple of 3, 4, and 6 is 12.

2

Distribute factors and combine like terms.

16x - 12 - 3x - 6 = 10x \implies 13x - 18 = 10x

Distributing

-3

into

(x + 2)

yields

-3x - 6

.

3

Isolate

x

on one side of the equation.

13x - 10x = 18 \implies 3x = 18 \implies x = 6

Subtract

10x

and add 18 to isolate the variable term.

4

Evaluate each given statement using

x = 6

.

Statement 1:

6

is a multiple of 3 (True). Statement 2:

2(6) - 5 = 7 > 6

(True). Statement 3:

6^2 - 5(6) = 6

(True). Statement 4: 6 is prime (False). Statement 5:

6 < 5

(False).

Test each option against the calculated solution

x = 6

.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 318Question →

A venue offers two types of event packages: Standard and Deluxe. The total cost of 3 Standard packages and 2 Deluxe packages is $410. The total cost of 2 Standard packages and 5 Deluxe packages is$ 640. What is the cost, in dollars, of 1 Deluxe package?

Show answer & explanation

Answer: 100

Answer

100

Setting up the linear system

3S + 2D = 410

and

2S + 5D = 640

allows us to multiply the equations by 2 and 3 respectively, obtaining

6S + 4D = 820

and

6S + 15D = 1920

. Subtracting the two equations eliminates

S

and gives

11D = 1100

, leading to

D = 100

.

Step-by-Step Solution

1

Define variables and establish the system of linear equations.

Let

S

represent the cost of a Standard package and

D

represent the cost of a Deluxe package.
Equation 1:

3S + 2D = 410

Equation 2:

2S + 5D = 640

Translating the scenario into mathematical equations forms a 2x2 system of linear equations.

2

Use elimination to eliminate variable

S

.

Multiply Equation 1 by 2:

6S + 4D = 820

Multiply Equation 2 by 3:

6S + 15D = 1920

Creating matching coefficients for

S

allows elimination by subtraction.

3

Subtract the transformed equations and solve for

D

.

(6S + 15D) - (6S + 4D) = 1920 - 820 \implies 11D = 1100 \implies D = 100

Subtracting cancels out

S

, leaving a single linear equation in terms of

D

.

Key Concept

Solving 2x2 Systems of Linear Equations via Elimination

Alternative Method

Use the substitution method: Solve for

S

in terms of

D

from the first equation (

S = \frac{410 - 2D}{3}

) and substitute this into the second equation (

2\left(\frac{410 - 2D}{3}\right) + 5D = 640

). Multiplying both sides by 3 yields

820 - 4D + 15D = 1920

, which simplifies to

11D = 1100

, so

D = 100

.

Estimated Time:1m 30s

Question 319Question →

In the $xy$ -plane, line $L_1$ passes through the points $(1, 1)$ and $(3, 5)$ . Line $L_2$ is defined by the equation $\frac{x}{4} + \frac{y}{7} = 2$ . If $(x, y)$ is the point of intersection of lines $L_1$ and $L_2$ , what is the value of $x + y$ ?

11

22

4

9.5

15

Show answer & explanation

Answer: 11

Answer

The value of

x + y

is 11.

The correct solution first determines the equation of the first line,

y = 2x - 1

, from its given points. Substituting this into the second line's equation

\frac{x}{4} + \frac{y}{7} = 2

and clearing fractions yields

x = 4

and

y = 7

. Adding these coordinates gives

4 + 7 = 11

.

Step-by-Step Solution

1

Determine the slope and equation of line

L_1

.

The slope

m = \frac{5 - 1}{3 - 1} = 2

. Using point-slope form with

(1, 1)

,

y - 1 = 2(x - 1)

, which simplifies to

y = 2x - 1

.

Two points uniquely define a line, allowing us to express

y

in terms of

x

.

2

Substitute the expression for

y

into the equation for line

L_2

.

\frac{x}{4} + \frac{2x - 1}{7} = 2

.

At the intersection point, both equations share the exact same

(x, y)

values.

3

Clear the denominators by multiplying the equation by the least common multiple, 28.

7x + 4(2x - 1) = 56 \implies 7x + 8x - 4 = 56 \implies 15x = 60 \implies x = 4

.

Eliminating fractions simplifies solving for

x

.

4

Calculate the value of

y

and find

x + y

.

y = 2(4) - 1 = 7

, so

x + y = 4 + 7 = 11

.

The question specifically asks for the sum of the intersection coordinates.

Key Concept

Systems of Linear Equations and Line Intersections

Estimated Time:1m 30s

Question 320Question →

If

a

,

b

, and

c

are pairwise distinct real numbers, which of the following expressions is equivalent to

\frac{a^3(b - c) + b^3(c - a) + c^3(a - b)}{(a - b)(b - c)(c - a)}?

-(a + b + c)

a + b + c

a^2 + b^2 + c^2

ab + bc + ca

-(ab + bc + ca)

Show answer & explanation

Answer: $-(a + b + c)$

Answer

-(a + b + c)

Factoring the numerator by using the Factor Theorem and cyclic symmetry reveals that

a^3(b - c) + b^3(c - a) + c^3(a - b) = -(a - b)(b - c)(c - a)(a + b + c)

. Dividing this by the denominator

(a - b)(b - c)(c - a)

cancels the pairwise difference terms, leaving

-(a + b + c)

.

Step-by-Step Solution

1

Analyze the numerator for polynomial factors using cyclic symmetry

Let

P(a, b, c) = a^3(b - c) + b^3(c - a) + c^3(a - b)

. If

a = b

, then

P(b, b, c) = b^3(b - c) + b^3(c - b) + 0 = 0

. By the Factor Theorem,

(a - b)

is a factor. By cyclic symmetry,

(b - c)

and

(c - a)

are also factors.

Identifying linear factors reduces the polynomial simplification problem.

2

Determine the degree and form of the remaining factor

P(a, b, c)

is a homogeneous polynomial of degree 4, while

(a - b)(b - c)(c - a)

has degree 3. Therefore, the remaining factor must be a homogeneous symmetric polynomial of degree 1, which takes the form

k(a + b + c)

for some constant

k

.

Homogeneous degree properties dictate the algebraic structure of the quotient.

3

Find the constant

k

by substituting test values

Let

a = 0

,

b = 1

, and

c = 2

. Evaluating

P(0, 1, 2) = 0 + 1^3(2 - 0) + 2^3(0 - 1) = 2 - 8 = -6

. The factor product gives

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

. Setting

2 \cdot k(0 + 1 + 2) = -6 \implies 6k = -6 \implies k = -1

.

Evaluating at convenient integer values determines the missing constant scalar.

4

Divide the factored numerator by the denominator

\frac{-(a - b)(b - c)(c - a)(a + b + c)}{(a - b)(b - c)(c - a)} = -(a + b + c)

.

Canceling common non-zero factors yields the simplified expression.

Key Concept

Factoring Cyclic Symmetric Polynomials

Algebra

Answer

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