# 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.

### 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:

−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.

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