There are a number of ways to solve simultaneous linear equations.
You can graph the two lines and find their point of intersection.
You can solve one equation for one of the variables in terms of the other, and then take the resulting expression involving the second variable and substitute the expression in place of the first variable in the second equation. Then solve the single variable equation that results. Use this result to substitute back into either of the original equations and solve for the other variable.
Find two constants such that if you multiply the first equation by the first constant and the second equation by the second constant you end up with one of the pair of variables having additive inverse coefficients. (In your example you don't have to multiply either equation by anything since the y variable already has a -1 coefficient in the first equation and a +1 coefficient in the second equation.). Then you add the two equations term by term which will eliminate one of the variables leaving you with a single variable equation to solve. Once you have the value of one of the variables, finding the other is trivial.
There is also Gauss-Jordan row reduction, but your example is to trivial to illustrate the process adequately. Generally Gauss-Jordan is a waste of time with 2X2 systems and is something you would use for 3X3 and larger systems.
You can use Cramer's Rule. I'll let you look this one up for yourself. Try PurpleMath.
John
My calculator said it, I believe it, that settles it