Solve linear equations including those with brackets, fractions and the unknown on both sides, form equations from contexts, and solve and represent linear inequalities on a number line.

What this dot point is asking

Solving linear equations and inequalities is a central CCEA Algebra skill. You must solve equations that contain brackets, fractions or the unknown on both sides, form equations from worded or geometric contexts, and solve inequalities and show their solutions on a number line. These appear across both tiers and on both the calculator and non-calculator work, and forming an equation from a context carries AO2 and AO3 reasoning marks.

Solving linear equations

A linear equation has the unknown to the first power only. The method is to keep the equation balanced: whatever you do to one side, do to the other, working towards isolating the unknown.

Start by expanding any brackets and clearing any fractions (multiply through by the denominator). Then collect the unknown on one side and the numbers on the other. Finally undo the operations attached to the unknown. For $5x + 4 = 19$, subtract 4 to get $5x = 15$, then divide by 5 to get $x = 3$.

When the unknown appears on both sides, subtract the smaller unknown term from both sides first, so $7x - 2 = 3x + 10$ becomes $4x - 2 = 10$, then $4x = 12$ and $x = 3$. Always substitute your answer back to check.

Equations with fractions

Fractions are cleared by multiplying every term by the denominator, or by the lowest common multiple of the denominators if there is more than one.

Forming equations from a context

Many marks come from translating words or a diagram into an equation. Let the unknown be a letter, write each piece of information as an algebraic statement, then solve. If the angles in a triangle are $x$, $2x$ and $x + 20$ degrees, they sum to $180$, so $4x + 20 = 180$, giving $x = 40$. Setting up the equation correctly is where the reasoning marks lie.

A worded example shows the pattern. Suppose a number is multiplied by 4 and 6 is added, giving 30. Let the number be $n$: the equation is $4n + 6 = 30$, so $4n = 24$ and $n = 6$. Always end by checking the answer against the original wording, since a value that does not fit the context (such as a negative length or a fraction of a person) signals an error in either the setup or the solving. Where two quantities are linked, such as one being three more than another, express both in terms of a single letter before forming the equation, because two unknowns in one equation cannot be solved on their own.

Inequalities

An inequality such as $2x + 1 < 9$ is solved with the same balancing steps as an equation, and here gives $x < 4$. The one extra rule is crucial.

To show a solution on a number line, use a filled circle when the value is included ($\le$ or $\ge$) and an open circle when it is excluded ($<$ or $>$), with an arrow in the direction of all the solutions.

Why this matters

Equations and inequalities are how mathematics models a condition and finds the value that satisfies it, from geometry and money to the constraints in optimisation. The balancing method here is the same one used for simultaneous and quadratic equations, so mastering it underpins the rest of the Algebra strand, and forming equations from context is exactly the AO3 problem solving CCEA rewards.