Simultaneous Equations — NSW Year 10 Advanced Maths
Two equations, two unknowns: substitution, elimination and graphs, lines meeting curves, and worded problems. Five subtopics and 100 practice questions with worked solutions, aligned to NESA outcomes MA5-EQU-P-02 and MA5-NLI-P-01.
Simultaneous Equations in NSW Year 10 Advanced
Simultaneous equations are pairs of equations that must both be true, and their solution is the point where their graphs meet. Year 10 Advanced students solve pairs of linear equations by substitution, by elimination and graphically, solve a line with a parabola, circle or hyperbola, such as \(y=x-3\) and \(xy=10\), and use simultaneous equations to solve problems.
The five subtopics below cover the substitution method, the elimination method and the graphical method, then non-linear simultaneous equations, and finish with worded problems. Simultaneous Equations is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Number and Algebra strand (NESA outcomes MA5‑EQU‑P‑02 and MA5‑NLI‑P‑01). The same methods are used in Year 11 Advanced for intersecting graphs and in many real-world models.
Subtopics
Simultaneous Equations: common questions
What are simultaneous equations?
They are two equations with two unknowns that are true at the same time. For \(y=3x\) and \(y=-2x+5\), the solution is \(x=1\), \(y=3\), the point where the two lines cross.
When should you use substitution rather than elimination?
Substitution is easiest when one equation already has \(x\) or \(y\) as the subject, such as \(y=3x-7\). Elimination is usually quicker when both equations are in the form \(ax+by=c\).
Can simultaneous equations have no solution?
Yes. If the two lines are parallel, with the same gradient but different intercepts, they never meet. For example, \(y=2x+1\) and \(4x-2y=6\) have no solution. If the equations describe the same line, there are infinitely many solutions.
How do you solve a line and a parabola simultaneously?
Substitute the line into the parabola to get a quadratic equation. For \(y=5x+6\) and \(y=x^2\): \(x^2-5x-6=0\), so \(x=6\) or \(x=-1\), giving the points \((6,36)\) and \((-1,1)\).
Which NESA outcomes cover Simultaneous Equations?
Solving linear simultaneous equations graphically, by substitution and by elimination, and worded problems, are MA5‑EQU‑P‑02. Finding where a line meets a parabola, hyperbola or circle is MA5‑NLI‑P‑01. Both are Path outcomes in Stage 5 of the NSW Mathematics K–10 Syllabus (2022).