Study Guide

Systems of Linear Equations - Solving Linear Systems

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.

  • More Vocabulary

    We know how to solve a system of two linear equations, not to mention what a system of equations actually looks like. We also know that a system of two linear equations has 0, 1, or infinitely many solutions. Now it's time to learn a bit more of the vocabulary that's used to describe the number of solutions to a system. While the phrases "a ton" and "more than its fair share" are moderately descriptive, there's more specific terminology we can use that will make your teachers happy, so let's dig in.

    A system is consistent if it has at least one solution. It also won't be accused of waffling if it runs for political office. A system with no solutions is called inconsistent. It can only successfully run for office in Chicago.

    Aww, don't look so sad, Chicagoans. It's funny because it's true.

    Sample Problem

    Is the system of equations

    consistent or inconsistent?

    If we graph these, we get lines that intersect exactly once, at (4, 5):

    Since these lines intersect, the system has a solution and is therefore consistent.

    Sample Problem

    Is the system of equations

    consistent or inconsistent?

    The second equation is what we find if we multiply the first equation by 2, which means these two equations are actually the same line and every point on the line is a solution. Since this system has infinitely many solutions, it certainly has at least one solution. Infinity is more than 1.

    You can trust us on this one...we've done the calculus. Anyway, the system is consistent.

    If a linear system of two equations is consistent, or has at least one solution, there are two possibilities: either the system has exactly one solution, or it has infinitely many. Either one line crosses the other at some point, or both are the same line. Hm. Sounds like the plot of Fight Club.

    We say a consistent linear system of two equations is dependent if the system has infinitely many solutions, and independent if the system has exactly one solution.

    Here's how you can remember these terms: you only need to listen to the song "Miss Independent" a total of one time in order for it to be stuck in your head for the next week. In fact, it's probably already stuck there now. Ah, the power of suggestion.

    So to sum up, a system of equations can be either consistent or inconsistent. If the system is consistent, it can be either dependent or independent.

    Dr. Math has a nice way to think about why the words consistent, inconsistent, dependent, and independent make sense from an English language point of view. Be careful, however. While he does use the title "Dr., " we do not recommend that you let him perform a kidney transplant on you.

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