Simultaneous equations: where two lines meet
Try values of x and y in two equations at once, see each one checked, and watch the guess move on a graph of both lines until it sits where they cross.
Core concept
The solution of two linear equations is the pair (x, y) that makes both true at once; on a graph it is the point where the two lines cross
Topic: Linear equations
An equation such as x + y = 10 has many solutions: (6, 4), (3, 7), (10, 0) and more. On a graph they make a straight line. A second equation, such as x − y = 2, makes another line. Only one pair (x, y) makes both true at once: the point where the two lines cross. Finding it is solving simultaneous equations. By elimination, add the two equations: 2x = 12, so x = 6 and then y = 4. By substitution, write y = 10 − x and put it into the other equation. Word problems become two equations too: if 7 tickets for adults (Rs 50) and children (Rs 20) cost Rs 260, then x + y = 7 and 50x + 20y = 260.
What you will learn
- Check whether a pair (x, y) satisfies two equations.
- Solve simultaneous equations by elimination or substitution.
- Explain the solution as the point where two lines cross.
- Write a word problem as two equations.
Before you start
- Know how to solve a linear equation in one variable.
Materials
- Graph paper and a pencil
Equipment
- None
Procedure
- Record the guess (5, 5) for x + y = 10 and x − y = 2.
- Change x and y until both equations are true, and record.
- Solve the ticket problem and record.
- Solve 2x + 3y = 18 and 3x − y = 5, and record.
How this simulation is simplified
- x and y are whole numbers from 0 to 10.