Introduction to Algebra: Variables, Expressions, and Perimeter — Practice Worksheet

1. True / False (1 pts)

Algebra uses variables to represent quantities that can change.

✓ Answer

True

2. True / False (1 pts)

In the term 9k, 9 is the variable.

✓ Answer

False

3. True / False (1 pts)

The perimeter of a square with side s is 3s.

✓ Answer

False

4. True / False (1 pts)

The distributive property allows you to expand a(b + c) into ab + ac.

✓ Answer

True

5. True / False (1 pts)

For all real numbers a and b, a + b = b + a.

✓ Answer

True

6. True / False (1 pts)

Variables must always be lowercase letters.

✓ Answer

False

7. True / False (1 pts)

Evaluating 4t − 1 at t = 0 gives 4.

✓ Answer

False

8. True / False (1 pts)

An algebraic expression may include more than one variable.

✓ Answer

True

9. True / False (1 pts)

In 6m + 2n, m and n are the literal coefficients.

✓ Answer

True

10. True / False (1 pts)

Two consecutive even integers differ by 1.

✓ Answer

False

11. Multiple Choice (2 pts)

In 12p, which identifies the variable and the numerical coefficient?

✓ Answer

B. Variable: p; Coefficient: 12

12. Multiple Choice (2 pts)

Which statement shows the commutative property of multiplication?

✓ Answer

C. ab = ba

13. Multiple Choice (2 pts)

If a = 5, what is the value of 3a + 4?

✓ Answer

B. 19

14. Multiple Choice (2 pts)

Which expression represents “twice a number decreased by seven”?

✓ Answer

B. 2x − 7

15. Multiple Choice (2 pts)

A rectangle has length 3x and width x + 2. Which expression equals its perimeter?

✓ Answer

C. 8x + 4

16. Multiple Choice (2 pts)

Which of the following is an algebraic expression?

✓ Answer

B. x/5 + 2

17. Multiple Choice (2 pts)

Which property lets you rewrite 7(n + 3) as 7n + 21?

✓ Answer

C. Distributive property

18. Multiple Choice (2 pts)

For an equilateral triangle with side y, the perimeter is:

✓ Answer

C. 3y

19. Multiple Choice (2 pts)

What is the constant term in 9 − 5q + 2?

✓ Answer

B. 11

20. Multiple Choice (2 pts)

If x = 1 and y = 4, what is 2x + 3y?

✓ Answer

B. 14

21. Fill in the Blank (2 pts)

A symbol that can take different values is called a _____.

✓ Answer

variable

22. Fill in the Blank (2 pts)

A number that stays the same in an expression is a _____.

✓ Answer

constant

23. Fill in the Blank (2 pts)

In 14z, 14 is the numerical _____.

✓ Answer

coefficient

24. Fill in the Blank (2 pts)

The rule a × (b + c) = ab + ac is the _____ property.

✓ Answer

distributive

25. Fill in the Blank (2 pts)

The perimeter of a polygon is the sum of the lengths of all its _____.

✓ Answer

sides

26. Fill in the Blank (2 pts)

The perimeter of a square with side s is _____.

✓ Answer

4s

27. Fill in the Blank (2 pts)

When x = 3, the value of 2x + 1 is _____.

✓ Answer

7

28. Fill in the Blank (2 pts)

Consecutive odd integers differ by _____.

✓ Answer

2

29. Fill in the Blank (2 pts)

An expression built from variables, constants, and operations is an algebraic _____.

✓ Answer

expression

30. Fill in the Blank (2 pts)

In 5p + 3q, p and q are the _____ coefficients.

✓ Answer

literal

31. Word Problem (5 pts)

A herb garden is planted in rows. The first row has 6 plants, and each following row has 4 more plants than the previous row. The number of plants in row r is given by 6 + 4(r − 1). How many plants are in row 8? Show your working.

✎ Worked Solution

Step 1: Substitute r = 8 into 6 + 4(r − 1): 6 + 4(8 − 1)

Step 2: Compute inside parentheses: 6 + 4(7) = 6 + 28

Step 3: Add: 6 + 28 = 34

Answer: 34 plants

32. Word Problem (5 pts)

A rectangle has length 3x + 2 and width x − 1. If x = 4, find its perimeter. Use P = 2(l + w).

✎ Worked Solution

Step 1: Compute length l = 3(4) + 2 = 12 + 2 = 14

Step 2: Compute width w = 4 − 1 = 3

Step 3: Find l + w = 14 + 3 = 17

Step 4: Perimeter P = 2(17) = 34

Answer: 34 units

33. Word Problem (5 pts)

Mira has n stickers. Leo has 2n + 3 stickers. If n = 12, how many stickers does Leo have?

✎ Worked Solution

Step 1: Substitute n = 12 into 2n + 3: 2(12) + 3

Step 2: Multiply and add: 24 + 3 = 27

Answer: 27 stickers

34. Word Problem (5 pts)

Evaluate 5(a + 2) − 3a when a = 4. Show each algebraic step.

✎ Worked Solution

Step 1: Expand using distributive property: 5a + 10 − 3a = 2a + 10

Step 2: Substitute a = 4: 2(4) + 10 = 8 + 10

Step 3: Add: 18

Answer: 18

35. Word Problem (5 pts)

Notebooks cost $2.25 each and pens cost $0.75 each. Write an expression for the total cost of 4 notebooks and 6 pens, then compute its value.

✎ Worked Solution

Step 1: Form the expression: Total = 4(2.25) + 6(0.75)

Step 2: Multiply: 4(2.25) = 9.00 and 6(0.75) = 4.50

Step 3: Add: 9.00 + 4.50 = 13.50

Answer: $13.50

36. Word Problem (5 pts)

An equilateral triangle has side length 12.3 cm. Find its perimeter.

✎ Worked Solution

Step 1: Use P = 3a for equilateral triangle with side a

Step 2: Substitute a = 12.3: P = 3(12.3) = 36.9

Answer: 36.9 cm

37. Word Problem (5 pts)

A square has side length y + 1 meters. If y = 6, find the perimeter of the square.

✎ Worked Solution

Step 1: Compute side: y + 1 = 6 + 1 = 7

Step 2: Perimeter of a square P = 4s = 4(7) = 28

Answer: 28 meters

38. Word Problem (5 pts)

The sum of two consecutive odd integers is 36. Find the two integers.

✎ Worked Solution

Step 1: Let the integers be n and n + 2

Step 2: Write equation: n + (n + 2) = 36 → 2n + 2 = 36

Step 3: Solve: 2n = 34 → n = 17; next is n + 2 = 19

Answer: 17 and 19

39. Word Problem (5 pts)

A row of n adjacent squares is built from matchsticks. The total number of matchsticks is M = 3n + 1. How many matchsticks are needed for 9 squares?

✎ Worked Solution

Step 1: Substitute n = 9 into M = 3n + 1

Step 2: Compute: M = 3(9) + 1 = 27 + 1 = 28

Answer: 28 matchsticks

40. Word Problem (5 pts)

A rectangle’s length is triple its width. If its perimeter is 64 units, find the rectangle’s width and length.

✎ Worked Solution

Step 1: Let width be w and length be 3w; perimeter P = 2(l + w) = 64

Step 2: Set up equation: 2(3w + w) = 64 → 2(4w) = 64 → 8w = 64

Step 3: Solve for w: w = 8; then l = 3w = 24

Answer: Width = 8 units, Length = 24 units