Over the last two quizzes, we’ve seen how to deal with systems involving two and three variables. We’ve also seen that systems sometimes fail to have a solution, or sometimes have “redundant” equations that lead to an infinite family of solutions. The natural question then becomes twofold: how can we solve general systems of equations, and how can we easily determine if a system has a unique solution?

In this and the next quiz, we’ll develop a method to do precisely that, called **Gaussian elimination**.

The previous problem illustrates a general process for solving systems:

1)Use an equation to eliminate a variable from the other equations. If there are $n$ equations in $n$ variables, this gives a system of $n - 1$ equations in $n - 1$ variables.

2)Repeat the process, using another equation to eliminate another variable from the new system, etc.

3)Eventually, the system “should” collapse to a 1-variable system, which in other words is the value of one of the variables. The remaining values then follow fairly easily.

For example, the previous problem showed how to reduce a 3-variable system to a 2-variable system. Repeating the process would reduce that 2-variable system to a 1-variable system, at which point we find out the value of $z$. This can be used to find $y$, then $x$, giving the full solution.

*one* of the equations has a given variable, we can always rearrange them so that equation is “on top.” But if none of the equations have a given variable, we have an issue.

For a 3-variable system, the algorithm says the following:

1)Eliminate $x$ from the second and third equations, using the first equation.

2)Eliminate $y$ from the third equation using the second equation.

3)Plug the value of $z$ into the second equation to get the value of $y$.

4)Plug the values of $y$ and $z$ into the first equation to get the value of $x$.

Which of these steps is the first that cannot be completed as described for the following system? $\begin{aligned} x + 2y + 3z &= 8 \\ 2x + 4y + 5z &= 15 \\ 3x + 6y - z &= 14 \end{aligned}$

**Gaussian elimination**, an algorithm to solve systems of equations. In the next quiz, we’ll take a deeper look at this algorithm, when it fails, and how we can use matrices to speed things up.