Add
To undo “− 3”, add 3 to the left and right sides.
Start here
x − 3=5
Add 3 to both sides
x − 3 + 3=5 + 3
Simplify
x=8
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GOOD HABITS, MADE SIMPLE.
It’s all about
balance.
The basics to help you find your way, one step at a time.
The idea in one sentence
An equation is a balanced scale. Whatever you do to one side, do to the other. The goal is to leave x on its own.
x + 3 = 8x = 8 − 3x = 5
Keep the balance
The same operation, with the same number, on both sides of the = sign.This lets you transform an equation without changing its solutions.
To leave x on its own, undo what surrounds it: undo subtraction with addition, and multiplication with division. Here is each move, line by line.
Choose the move you need. The first example is already open.
Addto remove − 3
Start here
x − 3=5
Add 3 to both sides
x − 3 + 3=5 + 3
Result
x=8
Soustraireto remove + 3
Start here
x + 3=8
Subtract 3 from both sides
x + 3 − 3=8 − 3
Result
x=5
Multiplierto remove ÷ 4
Start here
x ÷ 4=3
Multiply both sides by 4
(x ÷ 4) × 4=3 × 4
Result
x=12
Diviserto remove × 4
Start here
4x=12
Divide both sides by 4
4x ÷ 4=12 ÷ 4
Result
x=3
Subtract
To undo “+ 3”, subtract 3 from the left and right sides.
Start here
x + 3=8
Subtract 3 from both sides
x + 3 − 3=8 − 3
Simplify
x=5
Multiply
To undo “÷ 4”, multiply each side by 4.
Start here
x ÷ 4=3
Multiply each side by 4
(x ÷ 4) × 4=3 × 4
Simplify
x=12
Divide
To undo “× 4”, divide each side by 4.
Start here
4x=12
Divide each side by 4
4x ÷ 4=12 ÷ 4
Simplify
x=3
When multiplying or dividing, choose a non-zero number. Multiplying by zero would erase information; dividing by zero is impossible.
3x means 3 × x. The multiplication sign is omitted before a letter.
When there are two operations
The key move: first remove what is added or subtracted. Divide only at the end.
A number sits beside x
3x + 4 = 193x = 19 − 43x = 15x = 15 ÷ 3x = 5
x appears on both sides
5x − 7 = 2x + 85x − 2x = 8 + 73x = 15x = 5
Always check
Put x back into the original equation. If both sides give the same number, the answer is correct.
3 × 5 + 4=19✓
Two unknowns: systems of equations
Find x and y: a step-by-step guide
Two equations, one pair of answers. The most direct route is to eliminate one variable.
x + y = 7x − y = 1
- Eliminate y.Add the equations: +y and −y cancel out.
- Find x.This leaves 2x = 8, so x = 4.
- Find y.Put x into one equation: 4 + y = 7, so y = 3.
- Check both equations.4 + 3 = 7 et 4 − 3 = 1.
Nothing cancels yet?
Multiply an entire equation to create opposite coefficients, then add or subtract the equations.
x + 2y = 8 × 33x − y = 33x + 6y = 243x − y = 37y = 21y = 3, puis x = 2
Another route: substitute a variable
Isolate a variable in the first equation, then substitute it into the second. For example, y = 7 − x turns the second equation into x − (7 − x) = 1.
Watch out: always multiply the entire equation and always check both equations.
Going further
Open only the case that matches the equation you found.
ParenthesesDistribute a number
The factor multiplies every term inside the parentheses.
3(x + 4) = 3x + 12
−2(x − 5) = −2x + 10
−2(x − 5) = −2x + 10
A minus sign before parentheses changes the sign of every term inside. This is the most common mistake.
FractionsRemove denominators
Multiply the entire equation by the common denominator to eliminate the fractions.
x2 + x3 = 10
× 6 : 3x + 2x = 60
x = 12
× 6 : 3x + 2x = 60
x = 12
When one fraction equals another, you can cross-multiply: a/b = c/d gives ad = bc.
An unknown in the denominatorSpot excluded values
First note the excluded values: those that make a denominator zero. Then cross-multiply.
12x + 1 = 4 with x ≠ −1
12 = 4(x + 1)
x = 2
12 = 4(x + 1)
x = 2
If your solution is an excluded value, reject it.
Nested brackets and parenthesesStart from the inside
Always work from the inside out. Simplify the innermost parentheses, then the brackets around them, until you have a simple expression on each side.
2[3(x + 1) − 4] = 10
2[3x + 3 − 4] = 10
2[3x − 1] = 10
6x − 2 = 10, so x = 2
2[3x + 3 − 4] = 10
2[3x − 1] = 10
6x − 2 = 10, so x = 2
When a whole bracket is divided by a number, first multiply both sides by that divisor. This removes the division before you even expand the bracket.
Common mistakes to avoidMistakes to avoid
A minus sign before parentheses. It changes the sign of every term inside.
Changing only one side. This breaks the equality.
Checking an intermediate step. A mistake in step 2 can make every later step consistent with it, yet wrong. Always return to the original equation.
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