Linear Equations — How to Solve Step by Step

Grade: 7-9 | Topic: Algebra

What You Will Learn

By the end of this guide you will understand what a linear equation is, know the golden rule for solving equations, and be able to solve one-step, two-step, and multi-step linear equations confidently. You will also learn how to handle equations with variables on both sides, fractions, and word problems.

Theory

What is a linear equation?

A linear equation is an equation in which the highest power of the variable is 1. The word "linear" comes from "line" because these equations, when graphed, always produce a straight line.

The standard form of a linear equation in one variable is:

$ax + b = c$

where $a$, $b$, and $c$ are constants and $x$ is the variable we need to find.

Here are some examples:

Linear equation	Not a linear equation
$3x + 5 = 14$	$x^2 + 5 = 14$ (exponent is 2)
$\frac{x}{4} = 7$	$\sqrt{x} = 7$ (variable under a root)
$2(x - 3) = 10$	$\frac{1}{x} = 5$ (variable in denominator)

The golden rule — keep it balanced

An equation is like a balanced scale. Whatever you do to one side, you must do to the other side. This principle drives every step of solving:

$\text{If } \quad a = b \quad \text{then} \quad a + c = b + c$

$\text{If } \quad a = b \quad \text{then} \quad a \cdot c = b \cdot c \quad (c \neq 0)$

Inverse operations

To isolate the variable, use the inverse (opposite) operation to undo what is being done to it:

Operation on $x$	Inverse operation
Addition ($+ n$)	Subtraction ($- n$)
Subtraction ($- n$)	Addition ($+ n$)
Multiplication ($\times n$)	Division ($\div n$)
Division ($\div n$)	Multiplication ($\times n$)

Strategy for multi-step equations

For equations with more than one operation, follow this general order:

1. Simplify each side (distribute, combine like terms).
2. Move variable terms to one side using addition or subtraction.
3. Move constant terms to the other side.
4. Divide or multiply to isolate the variable.

Worked Examples

Example 1: One-step equation (easy)

Problem: Solve $x + 7 = 15$

Step 1: Subtract 7 from both sides to isolate $x$.

$x + 7 - 7 = 15 - 7$

$x = 8$

Answer: $x = 8$

Check: $8 + 7 = 15$ ✓

Example 2: Two-step equation (medium)

Problem: Solve $3x - 4 = 14$

Step 1: Add 4 to both sides to undo the subtraction.

$3x - 4 + 4 = 14 + 4$

$3x = 18$

Step 2: Divide both sides by 3.

$\frac{3x}{3} = \frac{18}{3}$

$x = 6$

Answer: $x = 6$

Check: $3(6) - 4 = 18 - 4 = 14$ ✓

Example 3: Equation with parentheses (medium)

Problem: Solve $2(x + 5) = 22$

Step 1: Distribute the 2.

$2x + 10 = 22$

Step 2: Subtract 10 from both sides.

$2x = 12$

Step 3: Divide both sides by 2.

$x = 6$

Answer: $x = 6$

Check: $2(6 + 5) = 2(11) = 22$ ✓

Example 4: Variables on both sides (challenging)

Problem: Solve $5x + 3 = 2x + 18$

Step 1: Subtract $2x$ from both sides to collect variable terms on the left.

$5x - 2x + 3 = 18$

$3x + 3 = 18$

Step 2: Subtract 3 from both sides.

$3x = 15$

Step 3: Divide both sides by 3.

$x = 5$

Answer: $x = 5$

Check: Left side: $5(5) + 3 = 28$. Right side: $2(5) + 18 = 28$. ✓

Example 5: Equation with fractions (challenging)

Problem: Solve $\frac{x}{3} + \frac{x}{6} = 5$

Step 1: Find the least common denominator (LCD). The LCD of 3 and 6 is 6. Multiply every term by 6.

$6 \cdot \frac{x}{3} + 6 \cdot \frac{x}{6} = 6 \cdot 5$

$2x + x = 30$

Step 2: Combine like terms.

$3x = 30$

Step 3: Divide both sides by 3.

$x = 10$

Answer: $x = 10$

Check: $\frac{10}{3} + \frac{10}{6} = \frac{20}{6} + \frac{10}{6} = \frac{30}{6} = 5$ ✓

Common Mistakes

Mistake 1: Forgetting to apply an operation to both sides

❌ $x + 5 = 12 \implies x = 12$ (subtracted 5 from the left but forgot the right)

✅ $x + 5 = 12 \implies x = 12 - 5 = 7$

Why this matters: If you only change one side, the equation is no longer balanced and your answer will be wrong.

Mistake 2: Distributing incorrectly

❌ $3(x + 4) = 3x + 4$ (forgot to multiply 4 by 3)

✅ $3(x + 4) = 3x + 12$

Why this matters: Every term inside the parentheses must be multiplied by the factor outside. Missing one term is one of the most common algebra errors.

Mistake 3: Sign errors when moving terms

❌ $4x - 7 = 9 \implies 4x = 9 - 7 = 2$ (kept the minus sign instead of changing it)

✅ $4x - 7 = 9 \implies 4x = 9 + 7 = 16$

Why this matters: Moving a term to the other side changes its sign. If you subtract 7 on the left, you add 7 on the right.

Practice Problems

Try these on your own before checking the answers:

1. Solve $x - 9 = 4$
2. Solve $5x + 2 = 27$
3. Solve $4(x - 3) = 20$
4. Solve $7x - 5 = 3x + 11$
5. Solve $\frac{2x}{5} + 3 = 7$

Click to see answers

1. $x = 13$ — Add 9 to both sides.
2. $x = 5$ — Subtract 2, then divide by 5: $5x = 25$, $x = 5$.
3. $x = 8$ — Distribute: $4x - 12 = 20$, add 12: $4x = 32$, divide by 4.
4. $x = 4$ — Subtract $3x$: $4x - 5 = 11$, add 5: $4x = 16$, divide by 4.
5. $x = 10$ — Subtract 3: $\frac{2x}{5} = 4$, multiply by 5: $2x = 20$, divide by 2.

Summary

• A linear equation has variables with exponent 1 and graphs as a straight line.
• The golden rule: whatever you do to one side, you must do to the other.
• Use inverse operations to undo addition, subtraction, multiplication, or division.
• For multi-step equations: simplify, collect variables, collect constants, then isolate the variable.
• Always check your answer by substituting it back into the original equation.

Related Topics

• Solving One-Step Equations
• Solving Two-Step Equations
• Solving Equations with Fractions
• Equation Word Problems
• Fractions — Complete Guide

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