solving for two equations at the same time means you want to find a value for x and a value for y that make both equations true. By true I mean when you plug in the values for x and y, you get the same number on both sides of the equal sign.
Using the process of elimination, we add or subtract the two equations and in doing so, one of the variables, either the x or the y will get eliminated. For this to happen, the coefficients of one of the variables have to be the same number or opposites meaning one negative and one positive but still the same number.
If you look at this question you will see that the coefficients for the y variable are the same but opposite sign so that is the variable we should eliminate. Because the signs are opposite, we add the equations because positive 9 plus negative 9 equals 0.
6x + 9y = 45
4x - 9y = -75
6x + 4x + 9y - 9y = 45 - 75
10x = -30
x = -3
Now that we know the value for x, we need to find the value for y.
To do this we go back to one of the original equations and plug in the value -3 where there is an x.
6x + 9y = 45
6(-3) + 9y = 45
-18 + 9y = 45
9y = 45 +18
9y = 63
y = 7
Therefore the solution is x = -3 and y = 7
You can also say the solution is the point (-3, 7)
In this example it was easy because the coefficients of y were 9 and -9 which were eliminated.
If the coefficients are not the same, you need to multiply one of the equations by a number so that the coefficients of one of the variables, either x or y, are made the same. When you multiply an equation, everything in the equation must be multiplied by that number.
