Year 11 General Linear Equations And Their Graphs

Solving Linear Equations With One Unknown

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An Intro to Solving Equations with Variables on Both Sides | Multi-Step Equations | Math with Mr. J Watch

Theory

A linear equation has the unknown raised only to the power of $1$ . To solve it, use inverse operations on both sides until the variable stands alone. This page covers the balance method, brackets, fractions, and multi-step examples.

A linear equation has the unknown variable raised only to the power of $1$ — no $x^{2}$ , no $\sqrt{x}$ , no $x$ in a denominator. To solve an equation means to find the value of the variable that makes the two sides equal.

The golden rule is the foundation of every method: whatever you do to one side of the equation, you must do to the other side. The equation stays balanced.

There are four operations that keep an equation balanced: add, subtract, multiply, and divide both sides by the same value (not zero for division). The aim each step is to undo whatever is being done to the variable, working in reverse order of operations until the variable stands alone.

An equation is a balance: whatever you do to one side, do to the other.

The four operations that preserve equality.

The distributive law for expanding brackets:

a (b + c) = a b + a c

a (b + c) = a b + a c

The four balance moves — each preserves the equation:

Operation	Effect
Add $k$ to both sides	$LHS + k = RHS + k$
Subtract $k$ from both sides	$LHS - k = RHS - k$
Multiply both sides by $k$	$k \cdot LHS = k \cdot RHS$
Divide both sides by $k$ ( $k \neq 0$ )	$\frac{LHS}{k} = \frac{RHS}{k}$

Clearing fractions. Multiply every term on both sides by the lowest common denominator (LCD). For $\frac{2 x}{3} - \frac{x}{5} = 14$ , the LCD is $15$ ; multiplying every term by $15$ clears the fractions in one move.

General strategy for any linear equation

Expand any brackets using the distributive law.
Clear fractions by multiplying every term by the LCD.
Collect variable terms on one side and number terms on the other.
Combine like terms on each side.
Divide by the coefficient of the variable to isolate it.
Check by substituting back into the original equation.

Sign flips. When you move a term across the equals sign, its sign flips: $x + 5 = 12$ becomes $x = 12 - 5$ . When you move a multiplier across, it becomes a divisor (and vice versa).

Example 1 — Two-Step

Solve

3 x - 7 = 11

Solution

Add $7$ to both sides, then divide by $3$ .

$3 x - 7$	$=$	$11$
$3 x$	$=$	$11 + 7$
$3 x$	$=$	$18$
$x$	$=$	$\frac{18}{3} = 6$

x = 6

Example 2 — Brackets

Solve

5 (x + 2) = 3 (x + 6)

Solution

Expand both brackets, then collect $x$ terms on one side.

$5 x + 10$	$=$	$3 x + 18$
$5 x - 3 x$	$=$	$18 - 10$
$2 x$	$=$	$8$
$x$	$=$	$4$

x = 4

Example 3 — Single Fraction

Solve

\frac{x}{4} - 3 = 1

Solution

Add $3$ to both sides, then multiply both sides by $4$ .

$\frac{x}{4}$	$=$	$1 + 3$
$\frac{x}{4}$	$=$	$4$
$x$	$=$	$4 \times 4$
$x$	$=$	$16$

x = 16

Example 4 — Multi-Term Fractions

Solve

\frac{2 x}{3} - \frac{x}{5} = 14

Solution

LCD is $15$ . Multiply every term by $15$ .

$15 \cdot \frac{2 x}{3} - 15 \cdot \frac{x}{5}$	$=$	$15 \times 14$
$5 (2 x) - 3 (x)$	$=$	$210$
$10 x - 3 x$	$=$	$210$
$7 x$	$=$	$210$
$x$	$=$	$30$

x = 30

Common pitfalls

Sign flip forgotten. When you move a term across the equals sign, its sign flips.

x + 5 = 12

becomes

x = 12 - 5

, not

x = 12 + 5

Distributing only one term. The number outside the brackets multiplies every term inside.

- 2 (x - 3) = - 2 x + 6

, not

- 2 x - 6

— the

- 2

hits both terms, and a negative times a negative is positive.

Forgetting a term when clearing fractions. Multiply every term by the LCD, including whole numbers.

\frac{x}{4} - 3 = 1

becomes

x - 12 = 4

after multiplying by

4

— the

- 3

and the

1

both get multiplied too.

Dividing only one term. When dividing by the coefficient, divide everything on both sides.

3 x = 18

divides both sides by

3

to give

x = 6

, not

x = 18

Frequently asked questions

What is a linear equation?

A linear equation has the unknown variable raised only to the power of 1. There are no x squared terms, no square roots of x, and no x in a denominator. Examples include 3x minus 7 equals 11 and 2 times x plus 5 equals 17.

What is the golden rule for solving equations?

Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced. You can add, subtract, multiply, or divide both sides by the same number (other than zero).

How do I solve an equation with brackets?

Expand the brackets first using the distributive law: a times open bracket b plus c close bracket equals ab plus ac. Be especially careful with negatives: minus 2 times open bracket x minus 3 close bracket equals minus 2x plus 6, because the minus 2 multiplies both terms inside.

How do I solve an equation with fractions?

Multiply every term on both sides by the lowest common denominator (LCD) to clear the fractions. For example, in two thirds of x minus x over five equals 14, the LCD is 15, so multiply every term by 15.

Why does the sign flip when I move a term across the equals sign?

Because moving a term across is actually shorthand for subtracting it from both sides. If x plus 5 equals 12, subtracting 5 from both sides gives x equals 12 minus 5. The plus 5 becomes minus 5 when it crosses over.

How do I check my answer?

Substitute your value back into the original equation. If both sides give the same number, your answer is correct. This is the most reliable way to catch arithmetic mistakes.

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More in Linear Equations And Their Graphs

Solving Linear Equations With One Unknown

📖 Theory

General strategy for any linear equation

Common pitfalls

Frequently asked questions

Theory