Linear Equations – One Variable, One Answer

A linear equation contains a variable raised to the power of 1 and produces a straight-line graph. Solving one means isolating the variable on one side of the equals sign by performing the same operations on both sides.

The word “linear” comes directly from the Latin linea (“line”), because every equation of this type graphs as a perfectly straight line — a direct visual consequence of the variable never being raised above the power of 1. Linear equations are the workhorse of everyday problem-solving: converting between temperature scales, working out a taxi fare from a fixed charge plus a per-kilometre rate, or calculating how many hours of work are needed to reach a savings goal are all one-variable linear equations in disguise.

What Makes an Equation Linear?

A linear equation has no powers higher than 1 on the variable. Every linear equation in one variable has exactly one solution. Examples: 3x + 5 = 11 and 7 - 2y = 1.

Solving One-Step Equations

Equation	Operation Needed	Solution
x + 8 = 13	Subtract 8 from both sides	x = 5
x - 4 = 9	Add 4 to both sides	x = 13
5x = 35	Divide both sides by 5	x = 7
x / 3 = 6	Multiply both sides by 3	x = 18

Solving Two-Step Equations

Solve 2x + 3 = 11.

Step 1 — subtract 3: 2x = 8. Step 2 — divide by 2: x = 4. Check: 2(4) + 3 = 11. Correct.

Solve 5y - 7 = 18.

Add 7: 5y = 25. Divide by 5: y = 5. Check: 5(5) - 7 = 18. Correct.

Variables on Both Sides

Solve 4x + 3 = x + 12.

Subtract x: 3x + 3 = 12. Subtract 3: 3x = 9. Divide by 3: x = 3. Check: 4(3)+3=15 and 3+12=15. Correct.

Solve 7 - 2m = 3m - 3.

Add 2m: 7 = 5m - 3. Add 3: 10 = 5m. Divide by 5: m = 2.

Equations with Fractions

Solve x/4 + 1 = 5.

Subtract 1: x/4 = 4. Multiply by 4: x = 16.

Solve (2x + 1) / 3 = 5.

Multiply by 3: 2x + 1 = 15. Subtract 1: 2x = 14. Divide by 2: x = 7.

Real-Life Application

A taxi costs a fixed fare of 3 plus 2 per kilometre. The total cost is 17. How far was the journey?

Set up and solve.

3 + 2d = 17. 2d = 14. d = 7 km.

Common Mistakes

Only doing an operation to one side of the equation.
Moving a term across the equals sign without changing its sign.
Dividing by a coefficient but forgetting to divide the entire side.

Key Takeaways

Isolate the variable by performing inverse operations equally on both sides.
Work backwards through the order of operations.
Always verify your answer by substituting back into the original equation.

Practice: Solve Linear Equations

Solve the Two-Step Equation