Solving Systems with Cramer’s Rule: Learn It 3

Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. As the order of the matrix increases to 3 × 3, however, there are many more calculations required.

When we calculate the determinant to be zero, Cramer’s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.

Consider a 3 × 3 system of equations.

A system of three linear equations in the variables x, y, and z, each with its own coefficients and constant on the right side.

[latex]x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D},z=\frac{{D}_{z}}{D},D\ne 0[/latex]

where

A diagram showing four determinants used in Cramer's Rule. The first is the original coefficient matrix. The next three replace, in turn, the x column, the y column, and the z column with the constants from the equations.

If we are writing the determinant [latex]{D}_{x}[/latex], we replace the [latex]x[/latex] column with the constant column. If we are writing the determinant [latex]{D}_{y}[/latex], we replace the [latex]y[/latex] column with the constant column. If we are writing the determinant [latex]{D}_{z}[/latex], we replace the [latex]z[/latex] column with the constant column. Always check the answer.

Find the solution to the given 3 × 3 system using Cramer’s Rule.

[latex]\begin{gathered}x+y-z=6\\ 3x - 2y+z=-5\\ x+3y - 2z=14\end{gathered}[/latex]

Solve the system of equations using Cramer’s Rule.

[latex]\begin{gathered}3x - 2y=4 \\ 6x - 4y=0\end{gathered}[/latex]

Solve the system with an infinite number of solutions.

[latex]\begin{gathered} x - 2y+3z=0\\ 3x+y - 2z=0 \\ 2x - 4y+6z=0 \end{gathered}[/latex]