multiply each equation by a number that produces opposite coefficient for x or y 4x +5y =7 3x-2y=-12

To produce opposite x terms:
-3(4x + 5y = 7) ⇒ -12x -15y = -21
4(3x - 2y = -12) ⇒ 12x - 8y = -48
so if you add the equations the x terms will cancel out

To produce opposite y terms:
2(4x + 5y = 7) ⇒ 8x + 10y = 14
5(3x - 2y = -12) ⇒ 15x -10y = -60
so if you add the equations now the y terms will cancel out

What you are doing is finding the LCM (least common multiple) of the coefficients for the terms you want to cancel out...
The LCM for 4 and 3 (coefficients of x) is 12 so you multiple each by what is required to get -12 and +12.
The LCM for 5 and 2 (coefficients of y) is 10 so you multiple each by what is required to get -10 and +10.