Expressions & Equations

Linear Expressions and Equations

Grade 7

✓By the end of this lesson students will be able to generate equivalent expressions by combining like terms and applying the Distributive Property.
✓By the end of this lesson students will be able to solve two-step linear equations using inverse operations.
✓By the end of this lesson students will be able to solve two-step linear inequalities using inverse operations, including reversing the inequality sign when necessary.
✓By the end of this lesson students will be able to interpret solutions to linear equations and inequalities.

Key concepts

Equivalent Expressions

Equivalent expressions are expressions that have the same value for all possible values of the variables. You can generate equivalent expressions by combining like terms and applying the Distributive Property.

Like Terms

Like terms are terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms, and 2y and -7y are like terms. Constants (numbers without variables) are also like terms, such as 4 and -10.

Distributive Property

The Distributive Property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. It allows you to remove parentheses in an expression.

a(b + c) = ab + ac

Linear Equation

A linear equation is a mathematical statement that shows two expressions are equal. It contains one or more variables, and the highest power of any variable is 1. The goal is to find the value(s) of the variable that make the equation true.

ax + b = c

Linear Inequality

A linear inequality is a mathematical statement that compares two expressions using an inequality symbol: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). The solution to an inequality is a set of values, not just a single value.

ax + b < c (or >, ≤, ≥)

Inverse Operations

Inverse operations are operations that undo each other. For example, addition is the inverse of subtraction, and multiplication is the inverse of division. We use inverse operations to isolate the variable when solving equations and inequalities.

Key facts to remember

1Equivalent expressions have the same value for any input of the variable(s).
2The Distributive Property is a(b + c) = ab + ac.
3Combine like terms by adding or subtracting their coefficients.
4To solve equations or inequalities, use inverse operations to isolate the variable.
5When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.
6The solution to an equation is a specific value, while the solution to an inequality is a range of values.
7Always perform operations on both sides of an equation or inequality to keep it balanced.

Worked examples

Example 1

Generate an equivalent expression for: 4(3x - 2) + 7x - 5

IApply the Distributive Property to 4(3x - 2): 4 × 3x - 4 × 2 = 12x - 8.

IIRewrite the expression: 12x - 8 + 7x - 5.

IIIIdentify like terms: (12x and 7x) and (-8 and -5).

IVCombine like terms: (12x + 7x) + (-8 - 5).

VPerform the addition and subtraction: 19x - 13.

Answer

19x - 13

Remember to distribute the number outside the parentheses to ALL terms inside.

Example 2

Solve the equation: 5x + 12 = 37

IIdentify the operations performed on the variable x: x is multiplied by 5, then 12 is added.

IITo isolate x, perform the inverse operations in reverse order. First, subtract 12 from both sides of the equation: 5x + 12 - 12 = 37 - 12.

IIISimplify both sides: 5x = 25.

IVNext, divide both sides by 5: 5x / 5 = 25 / 5.

VSimplify to find the value of x: x = 5.

VICheck your answer by substituting x = 5 back into the original equation: 5(5) + 12 = 25 + 12 = 37. Since 37 = 37, the solution is correct.

Answer

x = 5

Always perform the same operation on both sides of the equation to maintain balance.

Example 3

Solve the inequality: -3x + 8 ≤ 20

IIdentify the operations performed on the variable x: x is multiplied by -3, then 8 is added.

IITo isolate x, perform the inverse operations in reverse order. First, subtract 8 from both sides of the inequality: -3x + 8 - 8 ≤ 20 - 8.

IIISimplify both sides: -3x ≤ 12.

IVNext, divide both sides by -3. Remember to reverse the inequality sign when dividing by a negative number: -3x / -3 ≥ 12 / -3.

VSimplify to find the solution for x: x ≥ -4.

VICheck your answer by picking a value in the solution set (e.g., x = 0, since 0 ≥ -4): -3(0) + 8 ≤ 20 → 0 + 8 ≤ 20 → 8 ≤ 20. This is true. Now pick a value not in the solution set (e.g., x = -5, since -5 is not ≥ -4): -3(-5) + 8 ≤ 20 → 15 + 8 ≤ 20 → 23 ≤ 20. This is false, which confirms the inequality direction.

Answer

x ≥ -4

A critical step in solving inequalities: if you multiply or divide both sides by a negative number, you MUST reverse the inequality symbol.

Common mistakes

✗Forgetting to distribute the number outside the parentheses to all terms inside.
✗Incorrectly combining unlike terms (e.g., trying to combine '3x' and '5').
✗Not performing the same operation on both sides of an equation or inequality.
✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
✗Making arithmetic errors with positive and negative integers, especially when subtracting negative numbers.

Exam tips

★Show all your steps clearly. This helps you catch errors and allows for partial credit if your final answer is incorrect.
★Check your solution by substituting the value(s) back into the original equation or inequality to ensure it makes the statement true.
★Pay close attention to positive and negative signs throughout your calculations.
★Remember the order of operations (PEMDAS/GEMDAS) when simplifying expressions or checking solutions.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum