Linear Equations

Central to algebra is the use of letters, most frequently x and y, to represent numbers. Doing this allows us to deal systematically with quantities that are unknown yet of importance in an analysis. These unknown quantities are often referred to as variables, literally things that vary over a range of numbers or values. Sometimes the point is to express a quantitative procedure in a succinct way, and the use of letters merely constitutes convenient shorthand.
Example 1
A sales agent is paid a basic wage of £200 per week plus 10% commission on sales. The procedure for working out his/her total wage could be written as: 4
Total wage = 200 + 10% of sales
It is often more useful to abbreviate this by using letters. If y is used to represent the total wage and x to represent sales we can express the procedure as:
y = 200 + 0.1x
Using this we can find the total wage for a week when sales were £1200:
y = 200 + 0.1 * 1200 = 200 + 120 = £320
Example 2
The sales agent in Example1, currently receiving a wage of £200 plus 10% commission on sales, is offered the alternative of receiving 20% commission on sales with no basic wage. What is the minimum level of sales the agent would have to reach to make the alternative commission-only wage attractive?
The existing arrangement can be represented as:
y = 200 + 0.1x
where y represents the wage and x the sales.
The alternative can be expressed as:
y = 0 + 0.2x