Simultaneous Equations (Graphically)
Solve simultaneous linear equations by graphing for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). Solving two equations together means finding the single pair of \(x\) and \(y\) values that makes both true at once — on a graph, that is where the two lines cross.
You will draw both lines with graphing software or from a table of values, read off the point of intersection, check it in both equations, and spot when parallel lines give no solution — a handy skill for comparing two prices or plans in everyday life.
Theory
Simultaneous linear equations are two straight-line equations solved together. This Year 12 Standard 1 (NSW) guide shows how to solve them graphically — draw both lines, read and check the point of intersection, use a table of values, and decide whether a pair has one, none or infinitely many solutions.
Two equations such as \(y=2x-1\) and \(y=-x+8\) are simultaneous equations: the aim is to find the \(x\) and \(y\) that make both true at the same time.
Solving graphically means drawing both lines on the same axes (from a table of values or a graphing app). The point of intersection lies on both lines, so it is the solution. Check it by substituting into both equations.
Different gradients mean the lines cross once — one solution. The same gradient with a different \(y\)-intercept gives parallel lines — no solution. If both equations give the same line there are infinitely many solutions.
Write each line in gradient–intercept form:
At the solution both lines give the same \(y\)-value for the same \(x\):
| Gradients and intercepts | Graph | Number of solutions |
|---|---|---|
| Different gradients | Lines cross once | One solution |
| Same gradient, different intercepts | Parallel lines | No solution |
| Same gradient, same intercept | The same line | Infinitely many |
How to solve simultaneous equations graphically
- Rearrange each equation into the form \(y=mx+c\) if needed.
- Graph both lines on the same axes using a table of values or a graphing app.
- Read the point where the lines cross and write it as \((x,y)\).
- Check the point in both equations — each must give the same \(y\).
The lines cross at \((4,1)\). Check in both equations.
| \(y=x-3:\ 4-3\) | \(=\) | \(1\ \checkmark\) |
| \(y=-x+5:\ -4+5\) | \(=\) | \(1\ \checkmark\) |
The solution is \(x=4,\ y=1\), the point \((4,1)\).
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(y=2x+1\) | 1 | 3 | 5 | 7 |
| \(y=-x+7\) | 7 | 6 | 5 | 4 |
Look for the column where both rows match.
| \(\text{At } x=2:\ 2(2)+1\) | \(=\) | \(5\) |
| \(\text{and } -2+7\) | \(=\) | \(5\) |
Both lines pass through \((2,5)\), so the solution is \((2,5)\).
(a) \(y=3x+1\) and \(y=3x-2\)
(b) \(y=2x+3\) and \(3y=6x+9\)
Compare gradients and intercepts. (a) Both gradients are 3, intercepts 1 and \(-2\): the lines are parallel and never meet. (b) Divide \(3y=6x+9\) by 3.
| \(3y\) | \(=\) | \(6x+9\) |
| \(y\) | \(=\) | \(2x+3\) |
(b) is the same line as the first. So (a) no solution; (b) infinitely many solutions.
Substitute \(x=4\) into both equations.
| \(y=2x-5:\ 2(4)-5\) | \(=\) | \(3\ \checkmark\) |
| \(y=-x+6:\ -4+6\) | \(=\) | \(2\) |
| \(2\) | \(\ne\) | \(3\) |
\((4,3)\) is on the first line only, so Kai is not right — the solution must satisfy both equations.
Common pitfalls
Frequently asked questions
How do you solve simultaneous equations graphically?
Draw both straight lines on the same axes and read the point where they cross. That point lies on both lines, so its x and y values solve both equations at once.
How do you check the solution of simultaneous equations?
Substitute the x-value of the crossing point into both equations. If both give the same y-value as the point, the solution is correct.
How do you find the solution from a table of values?
Look for the x-value where both rows of the table have the same y-value. That pair of x and y values is the solution.
What does it mean if two lines are parallel?
Parallel lines have the same gradient but different y-intercepts. They never meet, so the pair of equations has no solution.
When do simultaneous equations have infinitely many solutions?
When both equations describe the same line. For example 3y equals 6x plus 9 simplifies to y equals 2x plus 3, so every point on that line is a solution.