Cramer's Rule at a Glance

Cramer's Rule has nothing to do with Seinfeld, guitars, or horrible movie divorces. It has everything to do with solving systems of equations using our uptight friends, determinants. That's how Cramer rolls.

Here are two random, innocent equations, ready to be solved by us:

3x – y = 5
-2x + 4y = 7

We want to learn the values of x and y, and we're tired of using our old-school substitution and elimination methods. Good news: there's a quick way to knock this out using matrices.

First, we're gonna find the determinant of the coefficient matrix:

D = (3)(4) – (-2)(-1) = 10

Next, we'll find the D of the x set, Dₓ. You take D as you know it and go from there:

Delete those x-values out of there:

Replace 'em with the constant values from the ends of both equations, and you've got D_x:

D_x = (5)(4) – (7)(-1) = 27

Finally, you can find x. That's because:

At last, our purpose becomes clear, right? Just like Hannibal, we love it when a plan comes together. Now, to find y, same bat procedure, same bat equations:

This time you delete the y-values out and—you guessed it—substitute in the constant values to get y.

And of course you know how to find y:

Now we know that:

There's only one leetle thing to remember. Remember how you learned that a fraction with a zero as a denominator is a sign of the apocalypse? If we figure out our coefficient determinant, D, is zero, that's the end of the road. We know that zero will be our denominator for both x and y, so there will either be no solutions or infinite solutions to the system of equations. That's because either the two lines represented by your two equations are parallel or the same freakin' line. What a mind-bender.

Just when we thought we knew all there is to know about determinants, it turns out that those cute little 2 × 2 matrices are just the tip of the iceberg.