Solving Linear Systems
Now, of course, comes the big question: if we're asked to solve a system of linear equations and not told how, which method do we use? Oh, sorry. No, we're not going to ask you to prom. We meant the other big question.
The short answer? It doesn't matter. Since you can check your answers by making sure your numbers really do satisfy both equations, it doesn't matter how you get to those answers. Within reason, of course. No asking a friend, using a calculator, or performing a seance to bring back Euclid so you can get his take on things.
The longer answer? With practice, you'll probably find you like some methods better than others. Go with the ones you like. Maybe they like you back. You should ask them to prom.
The longest (and most accurate) answer? Some systems of equations lend themselves to one method more easily than to other methods. If you can recognize such instances, it's going to save you a ton of work and heartache. The system
is great for graphing, because the numbers are small and tidy integers. We could graph this and probably wouldn't make any mistakes unless we suddenly have an unexpected episode of the "graph dizzies."
On the other hand, the system
is better suited for substitution. The numbers aren't quite tidy enough to easily graph, since the first equation has intercepts of 7 and . Because the y in the second equation has a coefficient of 1, we can solve the second equation for y and use substitution.
Finally, a system like
would be a royal pain to graph, and there's no easy way to solve either equation for an individual variable. Unless you're a masochistic prince, you would use addition/elimination for this system.
When in doubt, we recommend substitution or addition, since it can be hard to make accurate graphs for many systems. If, however, you have an irrepressible desire to draw, knock your socks off. Bring extra paper and a roll of Scotch tape.