Systems of linear equations

Two coffees and one cake cost 8 euros together, one coffee and one cake cost 5.50 euros. How much is the coffee? Puzzles like this have two unknowns, and one equation alone is not enough. Only two conditions together pin down the answer. That is exactly what systems of linear equations are about.

Two equations, two unknowns

In the simplest case a system of linear equations consists of two equations with the same two unknowns, usually x and y. A single equation like x + y = 10 has infinitely many solutions: 1 and 9, 4 and 6, 7 and 3, and so on. Once a second condition is added, say x − y = 2, only one pair remains that satisfies both: x = 6 and y = 4. So you are always looking for the pair of numbers that works in both equations at the same time.

Two lines, one intersection point

Each of the two equations describes a line in the coordinate system. All points on the line satisfy its equation. The solution of the system is the point that lies on both lines: the intersection point. For y = 2x and y = x + 3 that is the point (3, 6), because 2 · 3 = 6 and 3 + 3 = 6. Only when the two lines run parallel do they never meet, and then the system has no solution. That is the exception, though.

The substitution method

Instead of drawing you can calculate. If one equation is already solved for one unknown, say y = x + 3, you substitute this expression into the other equation. Then x + y = 11 becomes x + x + 3 = 11, so 2x + 3 = 11 and therefore x = 4. Plug that value back in and you get y = 7. Two unknowns are reduced to one, step by step. The café puzzle from the start cracks the same way: with the coffee price x and the cake price y, 2x + y = 8 and x + y = 5.50. The second equation gives y = 5.50 − x. Substituted into the first this becomes 2x + 5.50 − x = 8, so x + 5.50 = 8. The coffee therefore costs 2.50 euros and the cake 3 euros.