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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?

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.

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

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