How do you solve the following simultaneous equations? 4x-3y=18, 7x+5y=52

First, we need to try and eliminate one of the x or the y variables from both equations. Here, we are going to eliminate the y variable. In order to do this, we need to make the y coefficients equal in both equations, and we can do this by multiplying the first equation through by 5 and the second one through by 3, so that both y variables are equal to 15. We get:

20x-15y=90 21x+15y=156

If we add the two equations together, we will eliminate the y variable, as required, because -15+15=0. Hence, we are left with one equation in terms of x:

41x=246, so x=6.

We can substitute this result into the first equation to get: 120-15y=90, and we can solve this to get y=2.

Hence, the solution to the simultaneous equations is x=6 and y=2.