Question

Find the number of solutions for this system of equations.

3x + 5y = 20
-2x +5y = 20

How can I do this (and for other systems of equations) without taking all day? Thanks!

Answer 1

In this particular case, the problem tells you that both formulas equal 20. This means that 3x+5y = -2x +5y.

If you put the ys on one side and the xs on the other, the 5ys disappear.

3x + 5y -5y = -2x + 5y -5y

(what you do on one side, has to be done on the other).

That leaves 3x = -2x.

To get rid of the -2x on the right side, you would add 2x to both sides.

This gives

3x + 2x = -2x +2x

which simplifies to

5x = 0


The only number that works for x here is 0.

If you plug zero in for x in either of the original equations, you then get y = 4.

Answer 2

Obviously both equations cannot be true unless x = 0. That means, also, that
5y = 20.
One more step and you can get y.

That didn't take all day.

That is the only one solution.

Actually you were not even asked for what x and y are. You were asked how many solutions there are. Each equation is a straight line on a graph, but they have different slopes. At how many points can two such straight lines intersect?