Variable and Expressions Worksheets

Variables and Expressions are an important part of learning mathematics and computer science. These ideas are necessary for solving problems, writing equations, and developing algorithms.

Variables are placeholders for values that change, whereas expressions are combinations of variables, numbers, and operations that represent specific values or calculations.

Worksheets on variables and expressions assist students in grasping these concepts by providing practical exercises and problems to reinforce their learning.

What are Variable and Expressions?

Variables are symbols, often letters like x, y, or a, that represent unknown or changeable values in mathematical expressions and equations.

They allow us to generalize problems and create formulas that can be used to solve a wide range of problems.

Expressions are combinations of variables, numbers, and mathematical operations (like addition, subtraction, multiplication, and division) that represent a specific value.

Unlike equations, expressions do not have an equality sign and therefore do not assert that two quantities are equal.

Examples of Expressions

• 3x + 2 [x is the variable here]
• 4a − 5b + 7 [a and b is the variable here]
• 2(x² + 3x) [x is the variable here]

Concept and Formulas Related to Variable and Expressions

Some of the common concepts and formulas related to variable and expressions are:

Combining Like Terms

Like terms are terms that have the same variable(s) raised to the same power.

Formula: ax + bx = (a + b)x.

Example: 3x + 5x = 8x.

Distributive Property

This property allows you to multiply a single term by each term within parentheses.

Formula: a(b + c) = ab + ac.

Example: 2(x + 3) = 2x + 6.

Evaluating Expressions

To evaluate an expression, substitute the given values for the variables and perform the arithmetic operations.

Formula: If x = a, then f(x) = 3x + 2 becomes f(a) = 3a + 2.

Example: If x = 4, then 3x + 2 = 3(4) + 2 = 12 + 2 = 14.

Multiplying Expressions

Use the distributive property or FOIL method (First, Outer, Inner, Last) for binomials.

Formula: (a + b)(c + d) = ac + ad + bc + bd.

Example: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.

Factoring Expressions

Factoring is the process of writing an expression as a product of its factors.

Formula: ax² + bx + c = (px + q)(rx + s) (for quadratic expressions).

Example: x² + 5x + 6 = (x + 2)(x + 3).

Exponent Rules

• Multiplying powers with the same base: x^m × xⁿ = x^{m + n}.
• Dividing powers with the same base: x^m/xⁿ = x^{m - n}.
• Power of a power: (x^m)ⁿ = x^mn.

Example: x²×x³ = x^{2 + 3} = x⁵.

Variable and Expressions: Solved Examples

Problem 1: Simplify the expression 3x + 5 − 2x + 7.

Solution:

Combine like terms: 3x − 2x = x.

Combine constants: 5 + 7 = 12.

The simplified expression is x + 12.

Problem 2: Evaluate the expression 4a − 3b + 7 for a = 2 and b = − 1.

Solution:

Substitute the values: 4(2) − 3( − 1) + 7.

Simplify the expression: 8 + 3 + 7 = 18.

The value of the expression is 18.

Problem 3: Simplify the expression 2(3x + 4) − 5(x − 2).

Solution:

Distribute the 2: 2(3x + 4) = 6x + 8.

Distribute the - 5: − 5(x − 2) = − 5x + 10.

Combine like terms: 6x − 5x + 8 + 10 = x + 18.

The simplified expression is x + 18.

Problem 4: Evaluate 3(x + 2) − 4(2x − 1) when x = 3.

Solution:

Substitute the value: 3(3 + 2) − 4(2(3) − 1).

Simplify inside the parentheses: 3(5) − 4(6 − 1).

Multiply: 15 − 4(5) = 15 − 20.

Subtract: 15 − 20 = − 5.

The value of the expression is - 5.

Problem 5: If y = 2x² + 3x − 4, find y when x = − 1.

Solution:

Substitute x = − 1 into the expression: y = 2( − 1)² + 3( − 1) − 4.

Simplify: y = 2(1) − 3 − 4.

Perform the operations: y = 2 − 3 − 4 = − 5.

The value of y is - 5.

Worksheet: Variable and Expressions

Q1: Simplify: 7x + 3 − 4x + 5.

Q2: Evaluate: 2a + 3b − 5 for a = 4, b = − 2.

Q3: Simplify: 4(2x + 3) − 3(x − 2).

Q4: Evaluate: 5(x + 1) − 2(x − 3) for x = 0.

Q5: Simplify: − 2(3y − 4) + 7(y + 2).

Q6: Evaluate: 3x2 − 2x + 1 for x = 2.

Q7: Simplify: 9a + 2b − 3a + 4b.

Q8: Evaluate: 4p − 3q + 2 for p = 1, q = − 1.

Q9: Simplify: 6(2x + 1) − 4(3x − 5).

Q10: Evaluate: 2x2 + 5x − 3 for x = − 2.

Answer Key

Answer 1: 3x + 8

Answer 2: − 4

Answer 3: 5x + 18

Answer 4: 11

Answer 5: y + 18

Answer 6: 9

Answer 7: 6a + 6b

Answer 8: 9

Answer 9: 14

Answer 10: − 9

Read More,

• Variables in Maths
• Dependent and Independent Variable
• Algebraic Expressions
• Variables and Constant in Algebraic Expression