Linear equations: learn and practice

Who this lesson is for

This linear equations lesson is for grades 7–8. Grade 7 solves one-step equations such as 2x + 3 = 11. Grade 8 keeps that skill and moves on to slope and systems from the grade 8 hub. Both grades link to this single page.

Both sides stay equal

A linear equation says two expressions are equal, and the unknown appears to the first power only. In x + 7 = 15, x is the unknown. Your job is to find the number that makes the sentence true. You may add, subtract, multiply, or divide, but you must do the same thing to both sides. The two sides are like the two pans of a balance. If you take 7 from the left pan only, the balance tips and the sentence is no longer true.

What linear means here

Linear means the graph of the relationship is a straight line, and in these first equations it also means the unknown is not squared and not inside a product with another unknown. 3x = 18 is linear. x/4 = 5 is linear. x squared equals 9 is not a one-step linear equation of this kind, because the unknown is squared. Grade 7 practice on this page stays with one operation beside x. Later equations such as 2x + 3 = 11 need two inverse steps, and they use the same balance idea.

Worked example 1: undo addition

Solve x + 7 = 15. The 7 is added to x, so subtract 7 from both sides. x + 7 − 7 = 15 − 7. The left side is x. The right side is 8. So x = 8. Check in the original sentence: 8 + 7 = 15, which is true. If you had added 7 instead of subtracting, you would get x = 22, and 22 + 7 is not 15. The check catches that. The solution is the number 8, not the equation x = 8 written as a new puzzle.

Worked example 2: undo multiplication

Solve 3x = 18. The 3 is multiplied by x, so divide both sides by 3. 3x / 3 = 18 / 3. The left side is x. The right side is 6. So x = 6. Check: 3 × 6 = 18. Dividing by 3 is legal because 3 is not 0. You never divide both sides by 0. Notice that 3x means 3 times x, even though the times sign is hidden. Students sometimes subtract 3 and get x = 15. Then 3 × 15 is 45, not 18, so the check fails and the inverse was the wrong one.

Worked example 3: undo division

Solve x/4 = 5. The unknown is divided by 4, so multiply both sides by 4. 4 × (x/4) = 4 × 5. The left side is x. The right side is 20. So x = 20. Check: 20 / 4 = 5. Multiplying undoes the division because 4/4 is 1, and 1 times x is x. If the equation had been x/4 = 5/2, you would still multiply both sides by 4, and the right side would be 10. The move depends on what is happening to x, not on whether the other side looks friendly.

Read the operation beside x

Name the operation that joins the number to x before you calculate. Added means subtract that number. Subtracted means add that number. Multiplied means divide by that number. Divided means multiply by that number. Do the move on both sides, simplify, and substitute the result into the original equation. Then try a fresh one-step equation on the board.

Common questions about linear equations

Why do both sides change?

The equation is a balance. Changing one side only makes the two sides unequal, so it is a different sentence. The same move on both sides keeps the solution the same.

Is x = 8 the same as 8 = x?

Yes. Equality reads in both directions. Either form says the unknown and the number are the same. Most write-ups put the unknown on the left.

What if the check fails?

The inverse was the wrong operation, or a move was applied to one side only. Go back to the original equation and name the operation beside x again.