Simultaneous Linear Equations

What they are and how to work with them

Steve Sugden

Bond University

10 May 2011

Solving a linear equation

Tony bought two cans of the same soft drink and received change of $4 from $10.

What is the cost of one can of soft drink?

This problem may be expressed as a linear equation in one unknown.

If we let the cost of one can be $x then we have 2x + 4 = 10

To solve it we can use elementary algebra:

We just solved one equation in one unknown.

It is important to check this solution.

Checking solutions

There are at least two types of checking. For an abstract problem, you should always perform the …rst one; for a "word problem", you should do both. Here they are:

Check #1: Substitute the value found back into the original equation and verify that it "works".

Check #2: Interpret the solution in terms of the original word problem to see if it makes sense.

If any check fails then you need to do some detective work to …nd any ‡aws and …x them.

This takes practice.

Problem

Do these checks for the soft drink example.

What are simultaneous equations?

These are two (or more) equations in two (or more) unknowns and occur very often in practical problems.

Example

Two people bought quantities of the same two items. Anne bought 4 biros and 3 rulers for $23. Bob bought 3 biros and 5 rulers for $29. What does each item cost? This problem (a so-called "word problem") can be formulated as two simultaneous equations. We suppose a biro costs $b and a ruler costs $r. Then:

How to solve 2 simultaneous equations

Solving 2 simultaneous linear equations

To solve this pair of equations we need a strategy, beyond what we used for just one equation.

Strategy: try to transform it to one like we just solved, i.e., with just one unknown.

Note: this is a general principle of problem-solving: try to reduce a new problem to one which you already know how to solve.