Addition Property of Equality (version 2): If a = b and c = d, then a + c = b + d.
Since systems of equations involve to sets of things that are equal we may add the equations together to give one new equation. If we do this in a way that eliminates one of the variables, then we'll be able to solve for the remaining variable.
Solution: If we add these equations together we'll get a new equation that can be solved for x.
Now we can solve this and get x = 1. To get y we'll substitute into one of the original equations and solve.
So the solution is (1,1).
Solution: Adding these equations together won't eliminate a variable. To fix this problem we'll multiply the second equation by -2 to get the new system below.
Now we can add these together.
Solving this we get y = 1. Now we can find x.
So the solution is (-2,1).
Solution: We can multiply the first equation by 2 and the second equation by -3, so that x will be eliminated. This gives the new system below.
Now we can add and get the equation.
-23 y = -69
Solving this gives y = 3. Now we can find x.
So the solution is (2,3).
Solution: We can multiply the first equation by 2 and we get the following equations.
Adding these together we get 0 = 0 (or we can recognize that they are the same equation) and so there are infinite solutions.
Solution: We can multiply the first equation by 2 and we get the following
Adding these together we get 0 = 7, which is impossible, so there are no solutions.