Skip to main content

1. Simultaneous Linear Equations

A system of simultaneous linear equations is written:

a1x + b1y = c1
a2x + b2y = c2

Our aim in this chapter is to find values (x, y) which satisfy both equations.

First, we will test a solution to see what it means. In later sections, we will see how to find the solution.


Two students are working on a chemistry problem involving 2 variables, x and y. They obtain this system of 2 equations in 2 unknowns:

−3x + y = 1

6x − 3y = −4

One student gets the solution `x = 1`, `y = 4`, while the other student's answer is `x=1/3,\ y=2.`

Who is correct?


We can write the answers using coordinates like the following.

First student's answer: `(1, 4)`.

Second student's answer: `(1/3,2).`

If `(1, 4)` satisfies both equations, then it is the correct answer.

Test in first equation:

`−3(1) + 4 = −3 + 4 = 1` [OK]

Test in second equation:

`6(1) − 3(4) = 6 − 12 = −6` [Not OK, should be `−4`]

Since the first answer doesn't work in both equations, we conclude it is not the correct solution.

Let's now try the second student's solution.

If `(1/3,2)` satisfies both equations, then it is a solution of the system.

By substitution:

Test in first equation:

`−3(1/3) + 2 = −1 + 2 = 1` [OK]

Test in second equation:

`6(1/3) − 3(2)= 2 − 6 = −4` [OK]

So we conclude the second student had the correct solution, `(1/3,2)` to the set of simultaneous equations.

Get the Daily Math Tweet!
IntMath on Twitter

We revise straight lines before seeing how to solve this kind of simultaneous equation using a graph.


Search IntMath, blog and Forum

Search IntMath

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Algebra Lessons on DVD

Math videos by

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

See the Interactive Mathematics spam guarantee.