Solve a pair of linear simultaneous equations by elimination and substitution, solve a linear and a quadratic simultaneously, and interpret the solution as the intersection of two graphs (Higher tier for non-linear).

What this dot point is asking

Simultaneous equations find values that satisfy two equations at once. In the CCEA Algebra strand you must solve a pair of linear equations by elimination and by substitution, and at Higher tier solve a linear equation together with a quadratic. You must also interpret the solution graphically, as the point or points where two graphs cross. These questions reward clear, organised working and a check, and the linear-and-quadratic case is a reliable Higher-tier mark earner.

Elimination

Elimination removes one variable so you can solve for the other. Make the coefficients of one variable equal in size, then add the equations if those terms have opposite signs, or subtract if they have the same sign.

If the equations are $2x + 3y = 13$ and $2x + y = 7$, the $x$ coefficients already match, so subtract: $(3y - y) = (13 - 7)$ gives $2y = 6$ and $y = 3$. Substitute back into either equation: $2x + 3 = 7$, so $x = 2$. When coefficients do not match, multiply one or both equations first to make them match.

Substitution

Substitution rearranges one equation to make a variable the subject, then replaces that variable in the other equation. It is the natural choice when one equation is already in the form $y = \ldots$.

A linear and a quadratic together (Higher)

When one equation is quadratic, elimination does not work, so use substitution. Rearrange the linear equation to make one variable the subject, substitute into the quadratic, and you get a single quadratic to solve. Each solution of that quadratic gives an $x$ value, and you find the matching $y$ from the linear equation. There can be two solution pairs, one (a tangent), or none, which corresponds to a line cutting a curve twice, touching it, or missing it.

The graphical interpretation

Every equation in $x$ and $y$ is a graph, and a simultaneous solution is a point that lies on both graphs, that is, where they cross. Two straight lines cross at one point (unless parallel), so two linear equations have one solution. A line and a circle or parabola can cross at two points, giving two solution pairs. This is why a graph can be used to estimate or check simultaneous solutions.

Two parallel lines never meet, so the equations have no solution, and this shows up algebraically as the variables cancelling to leave a false statement such as $0 = 5$. If the two equations are actually the same line, every point on the line is a solution, and the algebra collapses to a true statement such as $0 = 0$. Recognising these special cases stops you wasting time hunting for a single answer that does not exist. For a line and a curve, the number of intersection points (two, one, or none) matches the number of real solutions of the quadratic that the substitution produces, which links directly to the discriminant.

Why this matters

Simultaneous equations model situations with two conditions to satisfy at once, such as mixing problems, pricing and the intersection of paths. The algebraic methods feed directly into solving quadratics and into coordinate geometry, while the graphical interpretation connects algebra to the straight-line and quadratic graph topics. CCEA expects organised working and a check, both of which protect method marks.