
Brian H. answered 06/15/19
Experienced Math, Science, & Web Design Teacher
If the two equations have opposite coefficients, when you add the equations together, both variables will cancel out. So first, let's make a = -1 and b = -2 and see what happens.
x + 2y = 3
-x +(-2y) = -9
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0 = -6
You end up with a false statement, which tells you the system has No solution!
Now let's multiply the first equation by 3, so the constant terms are opposites. Then we will make a and b opposite to the coefficients on that equation.
3x + 6y = 9
-3x + (-6y) = -9
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0 = 0
Now you get a true statement. This means the problem has infinitely many solutions.
Lastly, if one variable cancels, then the problem will have one solution. Let's use the first equation and change one of the coefficients in the second equation so it cancels.
x + 2y = 3
x + (-2y) = -9
--------------------
2x = -6
x = -3
plugging -3 in for x into either equation gives you y = 3
one solution exists and is the coordinate (-3, 3)