Given a system of linear equations, Cramer's Rule is a handy way to solve for just one of the variables without having to solve the whole system of equations. They don't usually teach Cramer's Rule this way, but this is actually supposed to be the point of the Rule: instead of solving the entire system of equations to make sure you get the value you need, you can instead use Cramer's to solve for just the one value that you need.
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Cramer's Rule tells us to form certain determinants and divide them in order to find variables' values. To see how Cramer's Rule works, let's apply it to the following system of equations:
2x + y + z = 3
x − y − z = 0
x + 2y + z = 0
We have the left-hand side of the system with the variables (that is, the "coefficient matrix") and the right-hand side with the answer values.
our system of equations, color-coded:
2x + 1y + 1z = 3
1x − 1y − 1z = 0
1x + 2y + 1z = 0
Let's name the determinant of the coefficient matrix of the above system "D":
coefficient matrix's determinant:
The column of "answer" values on the right-hand side of the "equals" signs in the system of equations above can be made into its own little matrix:
(Technically this is a "column vector", but you almost certainly won't need to know that terminology right now, if ever. However, if your instructor uses this term, the tall skinny one-column matrix above is what is meant.)
We'll take the coefficient determinant D and replace the first column of values (being the blue coefficients from the x-column of the original system) with the "answer" values. We'll call the result "Dx" (pronounced as "dee-sub-eks")
the determinant with x-values replaced:
As you can see, the red values from the answer column have replaced the original blue values from the x-variable terms in the original system of equations.
Similarly, determinants with the other two variables' values replaced are formed and named similarly; namely, the determinants Dy and Dz for this system are:
...where the red answer values have replaced the y-values in the middle green column, and:
...where the red answer values have replaced the z-values in the right-hand purple column.
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We can evaluate each of these determinants (using the method explained here). The results are these:
So, now that we have all these determinants and their values, what do we do with them?
Cramer's Rule says that we can find the value of a given variable by dividing that variable's determinant be the regular coefficient-determinant's value. That is, Cramer's Rule specifies this relationship:
Taking the determinant values that we derived from the system they'd given us and doing the relevant divisions, we get these values for the variables:
And that's all there is to Cramer's Rule. To find whichever variable you want (call it, say, "β" or "beta"), just evaluate the determinant quotient Dβ ÷ D.
(Please don't ask me to explain why this works. If you really want to know, there are loads of proofs online, like this and this. Otherwise, just trust me that determinants can work many kinds of magic.)
2x + y + z = 1
x − y + 4z = 0
x + 2y − 2z = 3
To solve only for z, I first form and find the value of the coefficient determinant.
Then I form Dz by replacing the third column of values with the answer column:
Then I form the quotient and simplify:
So then my answer is:
z = 2
The point of Cramer's Rule is that you don't have to solve the whole system to get the one value you need. This saved me a fair amount of time on some physics tests. I forget what we were working on (something with wires and currents, I think), but Cramer's Rule was so much faster than any other solution method for finding the one value I needed — and God knows I needed the extra time.
Don't let all the subscripts and stuff confuse you; the Rule is really pretty simple. You just pick the variable you want to solve for, replace that variable's column of values in the coefficient determinant with the answer-column's values, evaluate that determinant, and divide by the coefficient determinant. That's all there is to it.
You can't divide by zero, so what does this mean if the determinant by which you need to divide evaluates to zero? I won't go into the technicalities here, but "D = 0" means that the system of equations has no unique solution; instead, the system may be inconsistent (that is, it has no solution at all) or dependent (that is, it has an infinite solution, which may be expressed as a parametric solution such as "(a, a + 3, a − 4)"). In terms of Cramer's Rule, "D = 0" means that you'll have to use some other method (such as matrix row operations) to solve the system. If D = 0, you can not use Cramer's Rule.