Simultaneous equations involve two variables, each of whose value must be found.
The following method quickly eliminates one of the variables, so that the resulting equation can be solved normally.
3x + 2y = 7
2x - 3y = -4
3x + 2y = 7 (a)
2x - 3y = -4 (b)
in order to eliminate, we must have matching coefficients.
Multiply (a) by 2 and (b) by 3 so that we can eliminate the x terms.
6x + 4y = 14 (a)
6x - 9y = -12 (b)
We can eliminate x by subtracting (b) from (a)
This gives 4y - -9y = 14 - -12
So 13y = 26
y = 2
Substitute this y value into one of the equations, say (a) to get
3x + 4 = 7
3x = 3
x = 1
Check by putting this into the other equation (b) to get 2 x 1 - 3 x 2 = 2 - 6 = -4
Solution is x = 1, y = 2