Solve simultaneous linear equations in two unknowns by elimination and by substitution, solve them graphically, and form simultaneous equations from worded contexts.

What this dot point is asking

WJEC requires you to solve a pair of simultaneous linear equations in two unknowns, by elimination and by substitution, to solve them graphically, and to form simultaneous equations from worded contexts. "Simultaneous" means both equations must hold at the same time, so the solution is the single pair of values satisfying both. The skill draws on linear-equation solving and connects to straight line graphs through the point of intersection. It appears on both components.

What simultaneous means

A single linear equation in two unknowns has infinitely many solutions; you need two equations to pin down one pair of values. The solution is the values of $x$ and $y$ that make both equations true at once, which graphically is the single point where the two lines cross.

The elimination method

Elimination removes one variable by combining the equations.

The substitution method

Substitution works well when one equation already gives a variable in terms of the other. Rearrange one equation to make a variable the subject, substitute that expression into the other equation, solve the resulting single equation, then back-substitute. For $y = 2x - 1$ and $3x + y = 9$, replace $y$ to get $3x + (2x - 1) = 9$, so $5x = 10$, $x = 2$ and $y = 3$.

Graphical solution and forming equations

Plotting both lines and reading off their intersection gives the solution graphically, which is useful when an equation is awkward to manipulate, though it is only as accurate as the graph. WJEC also sets worded problems, such as "$3$ coffees and $2$ teas cost GBP 11, while $1$ coffee and $4$ teas cost GBP 12"; let $c$ and $t$ be the prices, write $3c + 2t = 11$ and $c + 4t = 12$, and solve. The algebra is identical once the two equations are formed.

Choosing elimination or substitution

Both methods always reach the same answer, so pick whichever the equations suit. Substitution is cleanest when one equation already has a variable as the subject, for example $y = 3x - 1$, because you can drop the expression straight into the other equation. Elimination is usually faster when both equations are in the form $ax + by = c$, since you only need to scale and add or subtract. With practice you will see at a glance which path involves less arithmetic, and choosing well reduces the chance of a sign slip.

Reading the worded structure

The hardest part of a worded simultaneous-equations problem is forming the two equations, not solving them. Look for two separate pieces of information that each link the same two unknowns: a total cost and a different combination, two people's ages now and in the future, or quantities and their combined mass. Define each unknown with a clear letter, write one equation per piece of information, and only then solve. WJEC awards marks for correctly forming the equations even before any solving, so writing them down clearly is worth real credit.

Why this matters

Simultaneous equations are how you solve problems with two interlocking unknowns, from pricing to mixtures to geometry. They are the algebraic counterpart of two lines crossing, linking directly to the straight line graph work, and at Higher tier the same elimination and substitution ideas extend to a linear and a quadratic equation. WJEC rewards a clean elimination or substitution, both values, and a check in the original equations.