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.