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Cramer's Rule

Cramer's Rule has nothing to do with Seinfeld, guitars, or horrible movie divorces. No, this has everything to do with solving square systems using our uptight friends, determinants. This is how Cramer rolls. Here are two random, innocent equations, ready to be figured out by us. We want to learn the values of x and y, so:

3xy = 5
-2x + 4y = 7

First, we are going to go with what we know and find the coefficient determinant:

D = (3)(4) – (-2)(-1) = 10

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

and delete those x values out of there:

add in the = values and you've got D:

Dx = (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 = values to get y.

And of course you know how to find y:

Looking for this?

(x, 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? In this case, since we're ultimately talking about lines (remember graphing a bajillion equations into lines?), in this case you can't get convenient (x, y) values because either the two lines represented by your two equations are parallel or the same freakin line. What a mind bender.

If we figure out our coefficient determinant is zero, that's the end of the road. We know that 0 will be our denominator for both x and y, so we can't get nice neat values.

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

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