Solving Simple Equations

An equation is a question written in symbols. "What number, multiplied by 2, gives 10?" becomes $2x = 10$ . Solving it means getting $x$ alone on one side — and the same three steps work every time, whatever the numbers.

Concept The three steps

Follow them in order and the variable ends up alone. Skipping a step is what makes an equation feel harder than it is.

Example A one-step equation

What number, multiplied by 2, gives 10?

Write it as an equation:

2x = 10

The variable is multiplied by 2, so divide both sides by 2.

x = \dfrac{10}{2} = 5

Check:

2 \times 5 = 10

, which is correct.

⟹ x = 5

Always substitute your answer back into the original equation. It takes seconds and catches nearly every mistake.

Example A two-step equation

What number, multiplied by 2 with 7 added, equals 15?

As an equation:

2x + 7 = 15

Step 1 — move the 7 across:

2x = 15 - 7 = 8

Step 2 — divide by the coefficient:

x = \dfrac{8}{2} = 4

Check:

2(4) + 7 = 8 + 7 = 15

, which is correct.

⟹ x = 4

A term changes sign as it crosses the equals sign: $+7$ on the left becomes $-7$ on the right. That is really just subtracting 7 from both sides at once.

Example Two more to try

3x = 12

— divide both sides by 3, giving

x = 4

x + 5 = 12

— subtract 5 from both sides, giving

x = 7

The operation you apply is always the opposite of what is being done to the variable. Multiplication is undone by division; addition is undone by subtraction.

Note Mistakes to avoid

Forgetting to change the sign of a term when it crosses the equals sign.

Applying an operation to one side only — both sides must stay balanced.

Dividing before the constants have been moved across.

Dividing by the wrong number — it must be the coefficient of the variable.

Skipping the check at the end.

Summary

Solving an equation means getting the variable alone on one side.
Collect like terms, move constants and variables to opposite sides, then divide by the coefficient.
A term changes sign when it crosses the equals sign, because you are really doing the same thing to both sides.
Undo each operation with its opposite: division for multiplication, subtraction for addition.
Always substitute the answer back to check it.