POP

G3.14 Solving Linear Equations

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01 — Explanation

How does the balancing method work?

Whatever you do to one side of an equation, you must do to the other — this keeps both sides equal while you isolate the unknown. Undo operations in the reverse order to how they were built (reverse BIDMAS): undo addition/subtraction before multiplication/division.

Example: $2x-3=7$ . Add 3 to both sides: $2x=10$ . Divide by 2: $x=5$ .

02 — Explanation

How do I solve an equation with brackets?

Expand the brackets first, then apply the balancing method.

Example: $2(x+3)=10 \;\Rightarrow\; 2x+6=10 \;\Rightarrow\; 2x=4 \;\Rightarrow\; x=2$ .

Common Misconception: Forgetting to multiply every term inside a bracket, especially with a negative outside it — $-2(x-3)$ expands to $-2x+6$ , not $-2x-6$ .

03 — Explanation

How do I solve an equation with the unknown on both sides?

Collect the $x$ -terms onto one side first (ideally eliminating the smaller one, to avoid a negative coefficient), then the constants, then divide.

Example: $3x+4=x+8$ .

Subtract $x$ from both sides: $2x+4=8$ .
Subtract 4: $2x=4$ .
Divide by 2: $x=2$ .

Exam Tip: Always move the smaller $x$ -term across, not the larger one — it keeps the remaining coefficient positive and avoids an easy sign slip in the final division.

04 — Explanation

How do I solve an equation with a fraction in it?

Multiply every term on both sides by the denominator to clear the fraction before doing anything else.

Example: $\dfrac{x}{2}+3=5$ . Multiply every term by 2: $x+6=10$ . Subtract 6: $x=4$ .

05 — Explanation

How do I check my solution is actually correct?

Substitute your answer back into the original equation (not a simplified version) and confirm both sides match.

Example: Check $x=2$ solves $3x+4=x+8$ : LHS $=3(2)+4=10$ ; RHS $=2+8=10$ . Both sides agree, so $x=2$ is correct.

Exam Tip: This check takes seconds and catches almost every slip — make it a habit on every equation question, especially ones with brackets or the unknown on both sides. If the two sides don't match, re-examine your working line by line rather than starting over from scratch — you'll usually spot exactly which step introduced the error.

06 — Explanation

How do I turn a word problem into an equation?

Define the unknown as $x$ , write an expression for each quantity described, and use the relationship given (total, perimeter, angle sum) to form the equation — then always check your final answer makes sense in context.

Example: A triangle has angles $a°$ , $(a+10)°$ , and $(a+20)°$ . Angles in a triangle sum to $180°$ :

a+(a+10)+(a+20)=180 \;\Rightarrow\; 3a+30=180 \;\Rightarrow\; 3a=150 \;\Rightarrow\; a=50

Common Misconception: Stopping once you've found $x$ (or $a$ ) without answering the actual question asked. If the question wants a side length or angle, substitute your solved value back into the original expression for that quantity — and always include the units.

07 — Worked example

Worked Example

A parallelogram has adjacent sides $(x+4)$ cm and $(2x-1)$ cm, and a perimeter of 24 cm. Find the length of the longer side.

Perimeter equation: $2(x+4)+2(2x-1)=24$ .
Expand: $2x+8+4x-2=24$ .
Collect like terms: $6x+6=24$ .
Solve: $6x=18$ , so $x=3$ .
Substitute back: side lengths are $x+4=7$ cm and $2x-1=5$ cm. The longer side is 7 cm.