Solve linear equations in one unknown, including those with brackets, fractions and the unknown on both sides, and form linear equations from worded and geometric contexts.

What this dot point is asking

OCR reference A7 asks you to solve linear equations in one unknown, including those with brackets, fractions and the unknown on both sides, and to form such equations from worded or geometric situations. Solving equations is the central skill of algebra, and forming them tests the AO3 problem-solving that OCR weights heavily. The work appears on every paper and tier, so the balance method must be second nature.

The balance method

An equation says two expressions are equal. Whatever you do to one side you must do to the other, which keeps the balance and lets you isolate the unknown.

So for $3x + 5 = 20$, subtract $5$ to get $3x = 15$, then divide by $3$ to get $x = 5$. Reading the equation as "multiply by $3$ then add $5$" and reversing it ("subtract $5$ then divide by $3$") is the reliable mental script.

Brackets and fractions

Brackets and fractions are cleared before collecting terms.

Expand brackets first: $2(x + 4) = 18$ becomes $2x + 8 = 18$, then $2x = 10$ and $x = 5$. For fractions, multiply the whole equation by the denominator: $\dfrac{x + 1}{3} = 4$ becomes $x + 1 = 12$, so $x = 11$. When two fractions appear, multiply through by the common denominator to clear both at once. For example, $\dfrac{x}{2} + \dfrac{x}{3} = 5$ has common denominator $6$, so multiplying through gives $3x + 2x = 30$, hence $5x = 30$ and $x = 6$. Multiplying every term, including any whole numbers, by the common denominator is essential; missing one term is a frequent mistake.

When a bracket is multiplied by a negative, watch the signs on every term inside it. The equation $3 - 2(x - 4) = 9$ expands to $3 - 2x + 8 = 9$, because $-2 \times (-4) = +8$, then $11 - 2x = 9$, so $-2x = -2$ and $x = 1$. Reading the negative sign as belonging to the whole bracket, not just the first term, prevents the classic error here.

The unknown on both sides

When the unknown appears on both sides, gather it on one side.

Collecting the unknowns on the side where the coefficient stays positive avoids a negative $x$ term, which is tidier but not essential.

Forming equations

Many marks come from translating a situation into an equation. Common triggers are perimeters, angle facts (a straight line is $180^\circ$, angles round a point sum to $360^\circ$, a triangle is $180^\circ$), and "I think of a number" problems. Define the unknown clearly, write the relationship as an equation, solve, and then answer the actual question asked, which is often a length or angle rather than $x$ itself. OCR rewards the explicit equation as method, so always write it down.

For instance, "a rectangle is $3$ cm longer than it is wide and has perimeter $26$ cm; find its width" becomes $2w + 2(w + 3) = 26$. Expanding gives $2w + 2w + 6 = 26$, so $4w = 20$ and $w = 5$ cm. The length is then $8$ cm. Because the perimeter formula and the worded relationship both feed into one equation, this kind of problem tests AO3 problem solving as well as AO1 technique, which is exactly why OCR sets it. Always state what your unknown represents at the start, so the marker can follow the reasoning and award the method marks.