Given the system of equations:
#-8x + 3y = -7; -3x -4y = 23#
We need to isolate the variables #x,y# so we can find the solution for each. Be very careful of SIGN changes.
I see #+3y# is an easy positive number to start with, so moving the #-8x# away from it will require changing the sign as it moves to the other side of the #=# sign:
#3y = 8x - 7#
To make the other equation similar to this new one line up the variables in the same order (changing signs over the #=# again):
#-4y = 3x + 23#
In this case we have a negative for the #y# value, but we will leave that alone to help us in a future step.
For now, we need to multiply both equations to result in the lowest common multiplier of the two #y# variables, which turns out to be #3 xx 4 = 12#.
#12y = 32x -28# when both sides are multiplied by #4#.
#-12y = 9x + 69# when both sides are multiplied by #3#.
Now we can add the two equations so that the #y# variable will disappear.
#12y = 32x - 28#
#-12y = 9x + 69#
#0 = 41x + 41; -41x = 41; x = -1#
if we substitute the value of #x = -1# into either #given# equation:
#-8(-1) +3y = -7#
#8 +3y = -7#
#3y = -15#
#y = -5#
To check the answers, substitute both values into the other #given# equation:
#-3(-1) -4(-5) = 23#
#3 + 20 = 23#
