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.