Solve the simultaneous equations: 3x+4y=19, x-6y= -23.

In order to solve these simultaneous equations we first want to eliminate one of the variables (x or y) and find a solution to the variable remaining (y or x) and then use this solution to find the second variable. To simplify this problem we want to make one of the variables in both equations equal, i.e. in this problem we can multiply the second equation by 3 to get 3x-18y= -69, rearranging this to get 3x on its own we get 3x= 18y-69. Rearranging the first equation to get 3x on its own yields 3x=19-4y. We can now equate these two equations as both are equal to 3x, hence 18y-69=19-4y, rearranging to solve for y gives 22y=88 and hence y=4. Substituting y=4 into one of the original equations to solve for x, (using the second equation) x= 6y-23=6*4-23=24-23=1. Hence the solutions are x=1 and y=4.