Equations

What is an Equation?

An equation is a mathematical statement showing that two expressions are equal, connected by an = sign.

3x + 5 = 11 is an equation. 3x + 5 alone is just an expression.

Equations are like balanced scales: both sides must always be equal. Solving an equation means finding the value of the unknown variable that makes both sides balance.

Key Vocabulary

Term	Meaning	Example
Variable	An unknown letter	x, y, a
Coefficient	Number multiplying the variable	3 in 3x
Constant	A fixed number	5 in 3x + 5
Solution	The value that satisfies the equation	x = 2

The Golden Rule

Whatever you do to one side of an equation, you must do to the other. This keeps the scale balanced.

If you add 3 to the left side, add 3 to the right. If you divide the left by 2, divide the right by 2.

Solving One-Step Equations

Use the inverse operation (the opposite operation) to isolate the variable.

Addition/Subtraction:

• x + 7 = 15 → subtract 7 from both sides → x = 8
• y - 4 = 9 → add 4 to both sides → y = 13

Multiplication/Division:

• 3x = 18 → divide both sides by 3 → x = 6
• x/5 = 4 → multiply both sides by 5 → x = 20

Solving Two-Step Equations

Work in reverse order of operations — undo addition/subtraction first, then multiplication/division.

Example: 2x + 5 = 13

1. Subtract 5 from both sides: 2x = 8
2. Divide both sides by 2: x = 4

Example: 3x - 7 = 11

1. Add 7 to both sides: 3x = 18
2. Divide both sides by 3: x = 6

Equations with Fractions

Multiply both sides by the denominator to clear the fraction, then solve.

Example: x/4 - 3 = 2

1. Add 3 to both sides: x/4 = 5
2. Multiply both sides by 4: x = 20

Example: x/3 + 2 = 7

1. Subtract 2: x/3 = 5
2. Multiply by 3: x = 15

Equations with Brackets

Expand the brackets first, then solve.

Example: 4(x - 3) = 20

1. Expand: 4x - 12 = 20
2. Add 12: 4x = 32
3. Divide by 4: x = 8

Equations with Variables on Both Sides

Collect all variable terms on one side and all numbers on the other.

Example: 5x - 4 = 3x + 8

1. Subtract 3x from both sides: 2x - 4 = 8
2. Add 4 to both sides: 2x = 12
3. Divide by 2: x = 6

Example: 3x + 2 = x + 10

1. Subtract x: 2x + 2 = 10
2. Subtract 2: 2x = 8
3. Divide by 2: x = 4

Checking Your Answer

Always substitute your answer back into the original equation to verify.

For 2x + 5 = 13 with x = 4:

• Left side: 2(4) + 5 = 8 + 5 = 13
• Right side: 13
• Both sides equal 13, so x = 4 is correct ✓

Common Mistakes

• Doing different operations to each side — always apply the same step to both sides.
• Forgetting to expand brackets before collecting terms.
• Wrong inverse operation — the inverse of + is -, and the inverse of × is ÷.
• Sign errors — take extra care when negatives are involved: subtracting a negative adds.
• Not checking the answer — always substitute back to confirm.

Tips and Tricks

• Write out every step clearly — do not try to do two steps at once.
• Circle the variable and ask: "What operation is being done to it, and what is the inverse?"
• For equations with fractions, multiplying through by the denominator immediately simplifies the problem.
• If the equation has brackets on both sides, expand both before collecting like terms.
• Remember: equations have solutions (a specific value); inequalities have solution sets (a range).