# At a Glance - Cramer's Rule

**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:

3*x* – *y* = 5

-2*x* + 4*y* = 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.

#### Example 1

Use Cramer's Rule to find D, _{y}x, and y in the following system of equations.3 |

#### Example 2

Use Cramer's Rule to find D, _{y}x, and y in the following system of equations.-2 |

#### Example 3

Use Cramer's Rule to find D, _{y}x, and y in the following system of equations.2 |