Algebra
Linear equations in two variables
Find solutions of an equation like 2x + y = 7 and read its graph as a straight line.
An equation like 2x + 5 = 11 has one letter and one answer. Put a second letter in — 2x + y = 11 — and something quite different happens. Now x = 1 with y = 9 works, and so does x = 3 with y = 5, and so does x = 0 with y = 11. A solution is no longer a number; it is a pair of numbers, and there are endlessly many of them.
Why it is called linear
Plot every solution pair as a point and they all land in a dead straight line. That is the whole reason for the name: the equation and the line are two ways of writing the same thing, and any point on the line is a solution, whether or not you worked it out.
Worked example
Find three solutions of x + y = 5.
Choose any value you like for x. Take x = 0. Then y must be 5, giving (0, 5).
You are allowed to choose, because it is the pair that has to fit, not either number on its own.
Try it together
Is (4, 1) a solution of 3x - 2y = 10?
Put the two numbers in, work the left side out, and see whether it lands on 10.
1.The x-coordinate is 4. What is 3 × 4?
The quickest two points to plot
To draw the line, find where it crosses each axis. Put x = 0 to get the y-axis crossing, and y = 0 to get the x-axis crossing. Two points are enough to fix a straight line, and these two are the least work.
Have a go
Have a go on your own: in 4x + y = 20, what is y when x = 2?
Hint: Four twos are eight, and the whole left side has to reach 20.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.