# 1. Simultaneous Linear Equations

A system of simultaneous linear equations is written:

a_{1}x+b_{1}y=c_{1}

a_{2}x+b_{2}y=c_{2}

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.

### Example

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

−3

x+y= 16

x− 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?

Answer

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.

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