GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Simultaneous equations

x + y = 10 has endless answers. Add a second clue, x − y = 4, and only one pair survives. Here's why.

Ride the line y = 3x − 2 until you hit the red line

-2-1012-8-4048xy(0, -2)

x: 0. y: -2

slide me

The red line is y = 3 − 2x: The point only ever rides the other line, y = 3x − 2, so every pair it shows makes that equation true. Now test each pair in y = 3 − 2x. At (0, −2) the red line needs y = 3 − 2 × 0 = 3, not −2, so this point is not on it. Slide until the point sits on the red line as well. That single pair is the only one that works in both equations.

Watch out: When two lines cross between grid lines, a graph can only give you a rough reading. Algebra gets you the exact pair.

Maths · Algebra

Same two lines, no graph paper

Step through it. Each line says what was done to both sides, and why.

GoalSolve y = 3x − 2 and y = 3 − 2x
1
y = 3x − 2 and y = 3 − 2x
start

These are the two lines from the graph. Both happen to start 'y = …', and there's a quicker route for that (it's coming up). Here we'll use elimination, because it works on any pair of linear equations once the terms are lined up.

2
3
4
5
6
7

Step 1 of 7

These are the two lines from the graph. Both happen to start 'y = …', and there's a quicker route for that (it's coming up). Here we'll use elimination, because it works on any pair of linear equations once the terms are lined up.

Which method?

Let the equations choose the method

Pick the description that matches your pair of equations and follow it to the move you make first.

What do the equations look like? → What do the coefficients do?

5 routes.

On Look at the form of the two equations. 3 branches to choose from.

Every route ends at the same pair of values. The form of the equations decides which route gets you there fastest.

Your turn

Substitution: fill in the missing steps

Solve x = 3y + 6 and x = 5y + 2.

  1. x = 3y + 6 and x = 5y + 2Both equations tell you what x is. So 3y + 6 and 5y + 2 are equal: they're both x.
  2. missing step
Which line is step 2?

Predict, then check

Think about what the two graphs look like before you commit.

How many pairs (x, y) make both y = 2x + 1 and y = 2x − 3 true?

Spot the slip

The other route, and where it goes wrong

Solve 3x − y = 2 and 2x + y = 3 by making the x coefficients match.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two equations, two unknowns, and the one pair of values that makes both true at once.

What you need to know

  • Solving simultaneous equations means finding the values that make both equations true at the same time.
  • One equation in two unknowns has infinitely many solutions, so you need two different equations to pin down one pair.
  • Elimination: subtract when a letter has the same coefficient in both equations, add when its coefficients are a zero pair, and multiply first when neither is true.
  • Substitution: when one equation has a letter as its subject, replace that letter in the other equation with its expression, in brackets.
  • On a graph, the solution is the point where the two lines cross.
  • Always check your pair in both original equations.

The big picture

An equation like 3x − y = 2 is true for endlessly many pairs of x and y. Its graph is a whole line of them. A second, different equation gives a second line, and when the two lines cross, the pair that fits both equations is the crossing point. You can find that pair by drawing the graphs, by elimination (adding or subtracting equations to remove a letter) or by substitution (replacing a letter with an expression). Whichever route you take, finish by checking the pair in both equations.

Key points

1The solution is a pair. Find both values, and say which is x and which is y.
2Line up x terms, y terms and numbers before eliminating. Do exactly the same thing to every term on both sides.
3Subtracting a negative term is adding it: 3y − (−2y) = 5y.
4Choose the method from the form: a letter already the subject suggests substitution, and ax + by = c suggests elimination.
5A graph reading can be approximate when the lines cross between grid lines. Algebra gives the exact answer.
6Parallel lines (same gradient, different intercepts) never meet, so there's no solution. Two equations for the same line have infinitely many solutions.
7For a straight line and a quadratic, substitute the linear expression into the quadratic, solve, then find each partner value. There can be two, one or no pairs.
8In a word problem, define each letter precisely, state the answer in context, and treat an impossible value (like a negative number of items) as a sign of an error.

Worked example

Problem

At a café, 2 teas and 3 cakes cost £7.20, and 3 teas and 1 cake cost £5.20. Find the cost of one tea and one cake.

⚠ Watch out

Finding one letter and stopping. The answer is a pair: once you have x, substitute it to find y, then check both values in the equation you didn't use.

🧠

Memory hook

One equation draws a whole line of answers. Add a second line, and the answer is where they cross.

✓

Check yourself

Why can't you find x and y from 2x + y = 10 alone? What does adding a second, different equation change? Think about the graph.

Flashcards

(16)
What does it mean to solve a pair of simultaneous equations?
Find the values of the unknowns that make both equations true at the same time.
How many solutions does one linear equation in x and y have on its own?
Infinitely many: every point on its line. You need a second, different equation to pin down one pair.
Two lines are drawn on the same axes. Where is the solution of their equations?
At the point where the lines cross. Its coordinates give x and y.
Why might a solution read from a graph be only approximate?
The lines may cross between grid lines, so the reading is an estimate. Algebra gives the exact values.
Elimination: a letter has the same coefficient in both equations (e.g. +4y and +4y). What do you do?
Subtract one equation from the other.
What is a zero pair, and what do you do with one?
Coefficients of equal size and opposite signs, like −3y and +3y. Add the equations to eliminate that letter.
Neither letter's coefficients match or make a zero pair. What now?
Multiply one or both equations by a suitable number so one letter's coefficients match or become a zero pair, then subtract or add.
Why must you do the same thing to every term when you add or subtract equations?
It keeps the equation balanced. Both sides of each equation are equal, so adding or subtracting equal amounts keeps them equal.
What is 4y − (−3y)?
7y. Subtracting a negative term is the same as adding it.
When is substitution the natural method?
When one equation already has a letter as its subject, like y = … or x = ….
Why do you put brackets round an expression when you substitute it?
So that anything multiplying or subtracting that letter applies to every term of the expression, e.g. 2y becomes 2(3x + 1), not 6x + 1.
Parallel lines: how many solutions?
None. Same gradient, different y-intercepts, so the lines never meet.
Both equations describe the same line. How many solutions?
Infinitely many: the lines meet at every point.
How do you solve a linear equation with a quadratic one?
Substitute the linear expression into the quadratic, solve the quadratic in one letter, then put each value into the linear equation to find its partner.
How many solution pairs can a line and a quadratic curve have?
Two, one or none: one for each point where the line meets the curve.
A word problem gives you a solution of −2 items. What does that tell you?
Something has gone wrong: a count can't be negative. Check the equations and the working.

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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