KS3 · Maths

Solving simultaneous equations graphically

Every line is a crowd of points obeying one rule. Draw two lines and ask: is there a point that obeys both rules at once? Look where they cross.

Two lines, one meeting point

-10123-5-3-113xy(1, -1)

x: 1. y: -1

Drag the point along y = 2x − 3. Wherever you stop, the point fits y = 2x − 3. Your job is to find the one spot that is on the red line as well.

The red line is y = 2 − ½x: Your point starts at (1, −1). On the red line, x = 1 gives y = 2 − ½ = 1.5, not −1, so this point is not on both lines. Keep sliding. At (2, 1): 2 × 2 − 3 = 1 and 2 − ½ × 2 = 1. Both equations agree there, so the solution is x = 2, y = 1.

Which equation does each point fit?

Put each point's x into each equation and see if you get its y. Tick every equation the point fits, or none at all.

  • A Fits y = 2x − 3
  • B Fits y = 2 − ½x
  1. (3, 3)
  2. (2, 1)
  3. (1, −1)
  4. (0, −3)
  5. (0, 2)
  6. (4, 0)
  7. (3, 1)

Same graph, two different questions

Solve 2x − 3 = 2 − ½xvsSolve y = 2x − 3 and y = 2 − ½x

Both columns use the lines y = 2x − 3 and y = 2 − ½x, which cross at (2, 1).

Focus

What the answer looks like

Solve 2x − 3 = 2 − ½x

One value: x = 2

Solve y = 2x − 3 and y = 2 − ½x

A pair of values: x = 2, y = 1

The insight

This is the difference people trip over. The single equation has only x in it, so it wants one number. The pair asks for a point, and a point always has two numbers.

Where you find it on the graph

Solve 2x − 3 = 2 − ½x

The x coordinate of the crossing

Solve y = 2x − 3 and y = 2 − ½x

The whole crossing point, (2, 1)

How to check it

Solve 2x − 3 = 2 − ½x

Put x = 2 into both sides: 2 × 2 − 3 = 1 and 2 − ½ × 2 = 1. Both sides are 1.

Solve y = 2x − 3 and y = 2 − ½x

Put x = 2 and y = 1 into each equation: 1 = 2 × 2 − 3 ✓ and 1 = 2 − ½ × 2 ✓

Another pair of lines

Solve 2x − 3 = 2 − ½x

y = 2x + 4 and y = 1 − x cross at (−1, 2), so 2x + 4 = 1 − x has the solution x = −1

Solve y = 2x − 3 and y = 2 − ½x

The simultaneous equations y = 2x + 4 and y = 1 − x have the solution x = −1, y = 2

Your turn to fill in the gaps

Solve the simultaneous equations y = 3x − 3 and y = 7 − 2x by drawing their graphs.

  1. Table for y = 3x − 3: when x = 0, 1, 2, y = −3, 0, 3.Pick a few x values and work out y each time.
  2. missing step
Which line is step 2?

Three lines, three crossings

The lines y = 2x − 3, y = 3 − x and y = ½x − 6 are drawn on one graph. They cross at three points. For each point, tick every line it lies on. (Substitute to check.)

(2, 1)
(−2, −7)
(6, −3)

Predict, then check

Equations like 2x + 4y = 24 are not written as y = …, so graphing software is a quick way to draw them.

Graphing software shows that 2x + 4y = 24 and 2y − x = 4 cross at (4, 4). It also shows that 2x + 4y = 16 and 2y − x = 4 cross at (2, 3). Now predict: what is the solution of 2x + 4y = 16 and 2x + 4y = 24?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Draw both lines, find where they cross, and read off x and y.

What you need to know

  • A point lies on a line when its coordinates fit the line's equation. For example, (3, 3) is on y = 2x − 3 because 2 × 3 − 3 = 3.
  • Each linear equation has infinitely many solutions, one for every point on its line.
  • You can draw a linear graph by working out coordinates and plotting them, or by using graphing software. Two or more lines can go on the same axes.
  • Two lines with different gradients cross at one point, called the point of intersection. It is the only point that fits both equations.
  • The solution to a pair of simultaneous equations is a value of x and a value of y. Check them by substituting into both equations.
  • Parallel lines never meet, so a pair of equations whose lines are parallel has no solution.

The big picture

A straight-line graph shows every pair of x and y values that fits its equation. Draw two lines on the same axes and they usually cross at one point. That point is the only one that fits both equations, so it is the solution of the simultaneous equations. For example, y = 2x − 3 and y = 2 − ½x cross at (2, 1), so the solution is x = 2, y = 1. Always give both values and check them in both equations. Parallel lines never cross, so their equations have no solution.

Key points

1Simultaneous equations are two equations that we want the same solution for.
2The point where their graphs cross is the only point that fits both equations, so it is the solution.
3Give both coordinates of the crossing as the solution, for example x = 2, y = 1. A point on just one of the lines is not a solution.
4The x coordinate of the crossing of y = A and y = B solves the single equation A = B. That needs only x.
5If a graph has three lines, each crossing solves only the pair of equations whose lines meet there.
6Graphs give approximate solutions when lines cross between grid lines. Substituting into both equations shows whether your reading is exact.

Worked example

Problem

Graphing software draws the lines 3x − 2y = 12 and 4x + 3y = −1 on the same axes. Use the graph to solve this pair of simultaneous equations.

⚠ Watch out

Stopping at x = 2 and forgetting y, or picking a point that sits on only one of the lines. The solution is the single point on both lines, written as x = … and y = …, and it must work in both equations.

🧠

Memory hook

Two lines, one handshake. Each line is a crowd of points, and the crossing is the one point in both crowds. A point always comes with two numbers, so shake hands with x AND y.

✓

Check yourself

Software shows 4x − 5y = −17 and 5x + 2y = 20 crossing at (2, 5). What is the solution, and how would you prove it fits both equations?

Flashcards

(13)
How can you tell if a point lies on a line?
Its coordinates fit the equation. Put x in and you get y. For example, (3, 3) is on y = 2x − 3 because 2 × 3 − 3 = 3.
How many points fit one linear equation?
Infinitely many. Every point on its line fits it.
What is the point of intersection?
The point where two lines cross.
Why is the point of intersection the solution of two simultaneous equations?
It is the only point that lies on both lines, so it is the only point that satisfies both equations.
What are simultaneous equations?
Two equations that we want the same solution for, so that both are true at once.
What must the solution to a pair of simultaneous equations include?
A value for x AND a value for y, not just one of them.
Two ways to draw the graph of a linear equation?
Work out coordinates in a table and plot them, or use graphing software.
How do you check a solution you have read from a graph?
Substitute the x and y values into both equations. Both must work.
The lines y = 2x + 4 and y = 1 − x cross at (−1, 2). What does the x coordinate tell you?
It solves the single equation 2x + 4 = 1 − x, so x = −1.
What happens with parallel lines?
They never cross, so their pair of equations has no solution.
When do two straight lines give an intersection?
When they have different gradients. Then they cross at exactly one point.
Lines y = 2x − 3, y = 3 − x and y = ½x − 6 cross at (6, −3). Which pair of equations does (6, −3) solve?
y = 3 − x and y = ½x − 6. It is not on y = 2x − 3, because 2 × 6 − 3 = 9.
Why do answers read from a graph sometimes need to be called approximate?
When the lines cross between grid lines, you can only estimate the coordinates. Checking in both equations shows whether a reading is exact.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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