Simultaneous Equations (Graphically)
Master simultaneous equations the graphical way for NSW Year 12 Mathematics Standard 2. In this topic you graph two straight lines on the same axes and read the point of intersection — the pair of values that solves both equations at once.
You will learn to solve a pair graphically, decide whether a pair has one solution, no solution (parallel lines) or infinitely many, and find the break-even point where revenue first covers cost — a core Standard 2 skill for comparing real-world plans and pricing.
Theory
Simultaneous linear equations are two straight-line equations solved together. This Year 12 Standard 2 (NSW) guide shows how to solve them graphically — graph both lines, read the point of intersection, decide how many solutions the pair has, and find the break-even point where cost meets revenue.
A pair of simultaneous linear equations is two straight-line equations that must both be true at the same time. Their solution is the pair of values \((x,y)\) that satisfies both.
Solving graphically means drawing both lines on the same axes. The point of intersection — where the lines cross — is the solution, because that point lies on both lines at once.
A pair can have one solution (lines cross), no solution (parallel lines), or infinitely many (the same line twice). In money problems the crossing point is the break-even point, where revenue equals cost.
Each line is written in gradient–intercept form:
At the solution the two lines share the same point, so their \(y\)-values are equal there:
How to solve a pair graphically
- Rearrange each equation into the form \(y=mx+c\) if it is not already.
- Graph both lines on the same axes (plot the \(y\)-intercept and one more point).
- Read the coordinates of the point where the lines cross.
- Check and state the solution \((x,y)\) in both equations; in a money problem, interpret it as the break-even point.
Read the crossing point, then check it in both equations.
| \((1,3)\) | \(\) | \(\text{crossing point}\) |
| \(3\) | \(=\) | \(1+2\ \checkmark\) |
| \(3\) | \(=\) | \(-1+4\ \checkmark\) |
| \(\therefore\ (x,y)\) | \(=\) | \((1,3)\) |
Both lines pass through the same point at \(x=2\).
| \(2(2)-1\) | \(=\) | \(3\) |
| \(2+1\) | \(=\) | \(3\) |
| \(\therefore\ (x,y)\) | \(=\) | \((2,3)\) |
Compare the gradients. Equal gradients mean parallel lines.
| \(m_1\) | \(=\) | \(3\) |
| \(m_2\) | \(=\) | \(3\) |
| \(2\) | \(\ne\) | \(-1\) |
Same gradient, different intercept — parallel, so no solution.
Break-even is where revenue equals cost, \(R=C\).
| \(4n\) | \(=\) | \(20+2n\) |
| \(2n\) | \(=\) | \(20\) |
| \(n\) | \(=\) | \(10\) |
| \(C\) | \(=\) | \(\$40\) |
Selling 10 candles gives revenue \(=\) cost \(=\$40\).
Common pitfalls
Frequently asked questions
How do you solve simultaneous equations graphically?
Graph both straight lines on the same axes and read off the coordinates of the point where they cross. That point is the solution, because it is the only pair of x and y values that lies on both lines.
What does the point of intersection tell you?
It gives the solution of the pair: the x-value and y-value that make both equations true at the same time. Always state both coordinates, for example x equals 2 and y equals 3.
What happens if the two lines are parallel?
Parallel lines have the same gradient and never cross, so the pair has no solution. If the two equations are actually the same line, every point lies on both and there are infinitely many solutions.
What is a break-even point?
It is where a revenue line meets a cost line on the graph. At that number of items, income equals cost, so there is no profit and no loss; beyond it the business starts to make a profit.
Can two straight lines have more than one solution?
No. Two different straight lines either cross once (one solution) or are parallel (no solution). The only way to get more than one solution is if the two equations describe the exact same line, giving infinitely many.