Cramer's rule for solving systems

What is Cramer’s Rule?

Cramer’s Rule is a simple rule that lets us use determinants to solve a system of equations. It tells us that we can solve for any variable in the system by calculating

$\frac{D_v}{D}$

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where $D_v$ is the determinant of the coefficient matrix, with the answer column values substituted into the column representing the variable for which we’re trying to solve, and where $D$ is the determinant of the coefficient matrix.

Which means that, if we want to find the value of $x$ , we need to find $D_x/D$ , and if we want to find the value of $y$ , we need to find $D_y/D$ . All that sounds tricky, but let’s look at an example to break it down.

How to use Cramer’s rule to solve systems of equations

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Solving for individual variables using Cramer’s Rule

Example

Solve for $x$ in the system.

$2x-3y=5$

$3x+12y=-8$

Because we’re looking for the value of $x$ , we want to find $D_x/D$ . We need to start with the coefficient matrix for the system,

The answer column matrix is the matrix of constants from the right side of the system,

$D_x$ is the determinant of the coefficient matrix with the answer column matrix substituted into the $x$ -column,

$D$ is the determinant of the coefficient matrix,

Putting these values together, Cramer’s Rule tells us that the value of $x$ in the system is

Let’s do another example where we use Cramer’s Rule to solve for $y$ .

If we want to find x, we need to find D_x/D, and if we want to find Y, we need to find D_y/D.

Example

Use Cramer’s Rule to solve for the value of $y$ that satisfies the system.

$3x-2y=7$

$5x-8y=21$

The coefficient matrix is

The answer column matrix is

Then $D_y$ is what we get when we substitute the answer column matrix into the second column of the coefficient matrix, and then take the determinant of the result.