x - y = 3 -2x + 2y = -6 What can be said about the system of equations? Does it have one solution, infinitely many solutions, no solution or 2 solutions.

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Answer 1 · 2017-12-04T00:57:19

Infinitely many

We have two equations:

#E1: x-y=3#
#E2: -2x+2y=-6#

Here's our choices:

If I can make #E1# be exactly #E2#, we have two expressions of the same line and so there are infinitely many solutions.
If I can make the #x# and #y# terms in #E1 and E2# the same but end up with different numbers they equal, the lines are parallel and therefore there are no solutions.
If I can't do either of those, then I have two different lines that are not parallel and so there will be a point of intersection somewhere.
There is no way to have two straight lines have two solutions (take two straws and see for yourself - unless you bend one, you can't get them to cross twice). When you start learning about graphs of curves (such as parabolas), then you'll start looking for two solutions.

To see what we can do, I'm going to multiply #E1# by #-2#:

#-2(x-y=3)=>-2x+2y=-6#

Here I've made #E1# be exactly #E2#, and so there are infinitely many solutions.