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

The Eduqas algebra content requires you to solve linear equations in one unknown, however they are dressed up: with brackets to expand, fractions to clear, or the unknown appearing on both sides. It also asks you to form an equation from a worded or geometric situation and then solve it. Linear equations are foundational, appearing on both components and at both tiers, and the skill of forming an equation is exactly where Eduqas tests AO2 and AO3 reasoning, so it carries marks beyond the routine solving.

The balance method

An equation is a statement that two sides are equal. Whatever you do to one side you must do to the other to keep the balance, and you undo the operations applied to the unknown in reverse order.

The reverse order matters: in $3x + 7$ the unknown was multiplied by $3$ then $7$ was added, so you undo the $+7$ first, then the $\times 3$.

Brackets and fractions

When an equation contains brackets, expand them before collecting terms. So $4(x + 3) = 20$ becomes $4x + 12 = 20$, then $4x = 8$ and $x = 2$.

When an equation contains fractions, clear them by multiplying every term by the lowest common denominator.

The unknown on both sides

When the unknown appears on both sides, collect the unknown terms on one side (usually the side that keeps the coefficient positive) and the numbers on the other. For $7x - 4 = 3x + 12$, subtract $3x$ from both sides to get $4x - 4 = 12$, add $4$ to get $4x = 16$, and divide to get $x = 4$.

Forming equations

Many marks come from turning a situation into an equation. A worded problem ("I think of a number, multiply it by $4$ and subtract $3$ to get $17$") becomes $4n - 3 = 17$. A geometric fact provides the equation: angles on a straight line sum to $180^\circ$, angles round a point to $360^\circ$, and the angles of a triangle to $180^\circ$. Set the relevant sum equal to the total, then solve. Because forming the equation is the reasoning step, always state the fact you are using ("angles in a triangle sum to $180^\circ$") so the method mark is secure.

A perimeter or money context works the same way. If a rectangle has length $x + 5$ and width $x$, and its perimeter is $34$ cm, then $2(x + 5) + 2x = 34$, which gives $4x + 10 = 34$, so $x = 6$ and the rectangle is $11$ cm by $6$ cm. The discipline is always the same: name the quantity with a letter, write the relationship as an equation, solve it, then translate the answer back into the context (and check it makes sense, since a negative length would signal an error).

Checking your solution

Substituting the answer back into the original equation is the fastest way to catch a slip and is worth doing every time. For $7x - 4 = 3x + 12$ with solution $x = 4$, the left side is $7(4) - 4 = 24$ and the right side is $3(4) + 12 = 24$, so the two sides match and the answer is confirmed. On the non-calculator Component 1 this self-check costs only a few seconds and protects an otherwise hard-won mark, which is why examiners' reports repeatedly recommend it.