Expressions and Equations: Master the Basics of Algebra

Question 1

Graph the linear equation y = -2x + 3. What are its slope and y-intercept?

Accepted Answer

Slope: -2, y-intercept: 3

Answer

Slope: 2, y-intercept: -3

Answer

Slope: -3, y-intercept: 2

Answer

Slope: 3, y-intercept: -2

Question 2

Graph the absolute value function |x - 2| + 1 and describe its key features.

Accepted Answer

A V-shaped graph with a vertex at (2, 1).

Answer

A U-shaped graph with a vertex at (2, 1).

Answer

A straight line with a slope of 1 and a y-intercept of 1.

Answer

A parabola with a vertex at (0, 1).

Question 3

A car rental company charges an initial fee of $25 plus $0.15 for each mile driven. Write an inequality to represent the total cost of renting the car for one day if you drive less than 150 miles.

Accepted Answer

0.15x + 25 < 150

Answer

0.15x - 25 < 150

Answer

25x + 0.15 < 150

Answer

25 - 0.15x < 150

Question 4

Which of the following is not a valid algebraic expression?

Accepted Answer

John's age

Answer

2x + 5

Answer

3 - x

Answer

x²

Question 5

What is the equation of the line with a gradient of -2 and a y-intercept of 3?

Accepted Answer

y = -2x + 3

Answer

y = 2x + 3

Answer

y = -3x + 2

Answer

y = 3x + 2

Question 6

Which system of equations has no solution?

Accepted Answer

y = 2x + 1
y = 2x + 3

Answer

y = 2x + 1
y = -x + 3

Answer

y = 2x + 1
y = -2x + 3

Answer

y = 2x + 1
y = -2x + 5

Question 7

For x = 2, which inequality is true?

Accepted Answer

2x - 6 ≥ 0

Answer

x - 3 < 0

Answer

x + 1 > 5

Answer

x^2 - 4 > 0

Question 8

What is the slope-intercept form of the equation of the line with a slope of 2 and a y-intercept of -3?

Accepted Answer

y = 2x - 3

Answer

y = -2x + 3

Answer

y = 3x - 2

Answer

y = -3x + 2

Question 9

Find the equation of the line passing through the points (2, 5) and (4, 9).

Accepted Answer

y = 2x + 1

Answer

y = x + 3

Answer

y = x - 3

Answer

y = -2x + 9

Question 10

A rectangle has an area of 24 square units and a length of 6 units. What is its width?

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 11

Which expression represents the perimeter of a rectangle with length x and width y?

Accepted Answer

2x + 2y

Answer

x + y

Answer

xy

Answer

x^2 + y^2

Question 12

Which of the following is the graph of the line y = 2x + 1?

Accepted Answer

A line with a slope of 2 and a y-intercept of 1

Answer

A line with a slope of 1 and a y-intercept of 2

Answer

A line with a slope of -2 and a y-intercept of 1

Answer

A line with a slope of 1 and a y-intercept of -2

Question 13

Which inequality represents the solution set of x is greater than or equal to 5?

Accepted Answer

x ≥ 5

Answer

x ≤ 5

Answer

x < 5

Answer

x > 5

Question 14

Which of the following is the equation of a line with a slope of -1 and a y-intercept of 3?

Accepted Answer

y = -x + 3

Answer

y = x + 3

Answer

y = -x - 3

Answer

y = x - 3

Question 15

Solve the equation: 3x - 5 = 10

Accepted Answer

x = 15

Answer

x = 5

Answer

x = -5

Answer

x = 1

Question 16

Which inequality represents the solution set of x greater than 5?

Accepted Answer

x > 5

Answer

x ≥ 5

Answer

x ≤ 5

Answer

x < 5

Question 17

Solve the system of equations: x + y = 5, x - y = 1

Accepted Answer

x = 3, y = 2

Answer

x = 2, y = 3

Answer

x = -3, y = -2

Answer

No solution

Question 18

Determine the equation of the line that has a slope of -3 and a y-intercept of 2.

Accepted Answer

y = -3x + 2

Answer

y = 2x - 3

Answer

y = 3x + 2

Answer

y = -2x + 3

Question 19

Which statement accurately describes the graph of the linear equation y = 2x - 3?

Accepted Answer

The slope is 2, and the y-intercept is -3.

Answer

The slope is -3, and the y-intercept is 2.

Answer

The graph passes through the points (0, 0) and (1, 2).

Answer

The graph is a horizontal line.

Question 20

Solve the equation: 3/4x - 2 = 6

Accepted Answer

x = 10

Answer

x = 4/3

Answer

x = 8

Question 21

A farmer has a rectangular field with a perimeter of 120 feet. The length of the field is double the width plus 10 feet. What is the area of the field?

Accepted Answer

600 sq. ft.

Answer

300 sq. ft.

Answer

900 sq. ft.

Answer

1200 sq. ft.

Question 22

Which graph shows the solution to the inequality y > -2x + 1?

Accepted Answer

Graph with the region above the line y = -2x + 1 shaded

Answer

Graph with the region below the line y = -2x + 1 shaded

Answer

Graph with the line y = -2x + 1 with no shading

Question 23

To make the graph of g(x) pass through the point (3, 5), which transformation should be applied to the linear function f(x) = 2x - 3?

Accepted Answer

Vertical translation 8 units up

Answer

Vertical stretch by a factor of 2

Answer

Horizontal translation 3 units to the right

Question 24

Analyze the graph of the linear function f(x) = 2x + 1. Which of the following statements correctly describes its properties?

Accepted Answer

Positive slope, domain: all real numbers

Answer

Negative slope, range: positive values only

Answer

Domain: positive values only, y-intercept: (0, -1)

Answer

Undefined domain, x-intercept: (1, 0)

Question 25

Consider the graph of the linear equation y = -2x + 3. What is the y-intercept of this graph, and what does it represent in the context of the equation?

Accepted Answer

The y-intercept is (0, 3). It shows the initial value of y when x is 0.

Answer

The y-intercept is (3, 0). It shows the initial value of x when y is 0.

Answer

The y-intercept is (-2, 3). It represents the slope of the line.

Answer

The y-intercept is (-3, 0). It represents the x-coordinate of the point where the line crosses the y-axis.

Question 26

Write the algebraic expression for 'the product of a number and 7 less than that number'

Accepted Answer

n(n-7)

Answer

n - 7

Answer

7n

Question 27

Prove the following expression is an identity: $$(2x + 3)(x - 5) = 2x^2 - 7x - 15$$

Accepted Answer

$$(2x + 3)(x - 5) = 2x^2 - 7x - 15$$

Answer

$$(2x + 3)(x - 5) = 2x^2 - 10x + 15$$

Answer

$$(2x + 3)(x - 5) = 2x^2 + 7x - 15$$

Answer

$$(2x + 3)(x - 5) = 2x^2 + x - 15$$

Question 28

A construction company charges $50 per hour for labor and a flat fee of $100 for materials. Which equation represents the total cost, y, in dollars, of hiring the construction company for x hours?

Accepted Answer

y = 100 + 50x

Answer

y = 50x - 100

Answer

y = 100 - 50x

Answer

y = 50 + 100

Question 29

Simplify the expression: 2(x + 3)

Accepted Answer

2x + 6

Answer

2x + 3

Answer

x + 6

Answer

6x + 3

Question 30

A car travels at a constant speed of 60 miles per hour. If it has traveled 240 miles, how long did it take?

Accepted Answer

4 hours

Answer

2 hours

Answer

3 hours

Answer

5 hours

Question 31

Factor the expression x² - 9 into its simplest form.

Accepted Answer

(x - 3)(x + 3)

Answer

x² - 9

Answer

(x - 9)(x + 9)

Question 32

Which of the following is equivalent to 2(x - 3)?

Accepted Answer

2x - 6

Answer

2x - 3

Answer

2x + 6

Answer

2x + 3

Question 33

Which of the following is the slope of the line passing through the points (2, 5) and (4, 9)?

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 34

Find the inequality that represents the solution set of x < 5

Accepted Answer

-∞ < x < 5

Answer

x ≤ 5

Answer

-5 < x < ∞

Answer

x > 5

Question 35

Describe the graph of the linear equation y = 2x + 1

Accepted Answer

A straight line with a y-intercept of 1 and a slope of 2

Answer

A parabola with a vertex at (0, 1)

Answer

A straight line with a y-intercept of 0 and a slope of 1

Question 36

Find the perimeter of a rectangle with a length of x and a width of 5

Accepted Answer

2x + 10

Answer

x + 5

Answer

x - 5

Answer

2x - 10

Question 37

What value of x satisfies the equation x^2 - 4 = 0?

Accepted Answer

±2

Answer

1

Answer

±4

Answer

0

Question 38

Simplify the expression: (2x^2y)^3

Accepted Answer

8x^6y^3

Answer

8x^5y^2

Answer

2x^6y^3

Answer

6x^5y^2

Question 39

Find the inequality that represents the solution set of -3 ≤ x ≤ 5

Accepted Answer

-3 ≤ x ≤ 5

Answer

x ≥ -3 or x ≤ 5

Answer

-3 ≤ x < 5

Answer

-3 < x < 5

Question 40

Simplify the algebraic expression: 2x + 3 - 5x

Accepted Answer

-3x + 3

Answer

3x + 3

Answer

5x - 3

Answer

-3x - 3

Question 41

Factor the quadratic expression: x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 9)(x - 9)

Answer

(x - 3)^2

Question 42

Apply the distributive property to the expression: 3(2x - 5)

Accepted Answer

6x - 15

Answer

6x + 15

Answer

3x - 15

Answer

3x + 15

Question 43

Simplify the expression: 2(x + 3) - 4.

Accepted Answer

2x + 2

Answer

x + 6

Answer

2x - 2

Question 44

Translate the following problem into an algebraic equation: A rectangular garden has a length of 12 feet and a width of 8 feet. What is the area of the garden?

Accepted Answer

Area = Length × Width = 12 × 8

Answer

Area = Length - Width = 12 - 8

Answer

Area = Length + Width = 12 + 8

Answer

Area = Length ÷ Width = 12 ÷ 8

Question 45

Solve for y: 2x + 3y = 12, given that x = 2.

Accepted Answer

4

Answer

6

Answer

2

Question 46

Which expression is equivalent to 3x + 4y - 5x + 2y?

Accepted Answer

-2x + 6y

Answer

8x + 6y

Answer

15xy

Answer

6x - 2y

Question 47

Solve for x: 2(x - 3) = 10

Accepted Answer

8

Answer

5

Answer

10

Answer

15

Question 48

Which inequality represents the statement: The sum of twice a number and 5 is greater than 15?

Accepted Answer

2x + 5 > 15

Answer

2x - 5 > 15

Answer

2x - 5 < 15

Answer

2x + 5 < 15

Question 49

The equation y = 2x + 3 has a slope of:

Accepted Answer

2

Answer

-2

Answer

3

Answer

-3

Question 50

What is the y-intercept of the line y = -x + 5?

Accepted Answer

5

Answer

-1

Answer

1

Answer

-5

Question 51

Which of the following is the graph of the inequality x - y > 2?

Accepted Answer

Image of a shaded half-plane above the line x - y = 2

Answer

Image of a shaded line x - y = 2

Answer

Image of a shaded half-plane below the line x - y = 2

Question 52

A rectangular garden is 12 feet long and 8 feet wide. If the length is increased by x feet, what is the new area of the garden?

Accepted Answer

12(8 + x)

Answer

12x + 8

Answer

8(12 + x)

Answer

12(8 - x)

Question 53

Which of the following is the expression for the perimeter of a square with side length s?

Accepted Answer

4s

Answer

s^2

Answer

4s^2

Answer

2s

Question 54

A circle has a diameter of 10 cm. What is the circumference of the circle?

Accepted Answer

10π cm

Answer

5π cm

Answer

2π cm

Answer

20π cm