Determinants can be used to solve a linear system of equations using Cramer’s Rule.

Cramer’s Rule for Two Equations in Two Variables

Given the system

This system has the unique solution

where

When solving a system of equations using Cramer’s Rule, remember the following:

1. Three different determinants are used to find x and y. The determinants in the denominators are identical.

2. The elements of D, the determinant in the denominator, are the coefficients of the variables in the system; coefficients of x in the first column and coefficients of y in the second column.

3. , the determinant in the numerator of x, is obtained by replacing the x-coefficients, , in D with the constants from the right sides of the equations, .

4. , the determinant in the numerator for y, is obtained by replacing the y-coefficients, , in D with the constants from the right side of the equation, .

Example. Use Cramer’s Rule to solve the system:

5x – 4y = 2

6x – 5y = 1

Solution. We begin by setting up and evaluating the three determinants :

From Cramer’s Rule, we have

The solution is (6,7).

Cramer’s Rule does not apply if D = 0. When D = 0, the system is either inconsistent or dependent. Another method must be used to solve it.

Example. Solve the system:

3x + 6y = -1

2x + 4y = 3

Solution. We begin by finding D:

Since D = 0, Cramer’s Rule does not apply. We will use elimination to solve the system.

3x + 6y = -1 2x + 4y = 3
2(3x + 6y) = 2(-1) -3(2x + 4y) = -3(3)	Multiply both sides of equation 1 by 2 and both sides of equation 2 by –3 to eliminate x
6x + 12y = -2 -6x – 12y = -9	Simplify
0 = -11	Add the equations

The false statement, 0 = -11, indicates that the system is inconsistent and has no solution.

Cramer’s Rule can be generalized to systems of linear equations with more than two variables. Suppose we are given a system with the determinant of the coefficient matrix D. Let denote the determinant of the matrix obtained by replacing the column containing the coefficients of "n" with the constants from the right sides of the equations. Then we have the following result:

If a linear system of equations with variables x, y, z, . . . has a unique solution given by the formulas

Example. Use Cramer’s Rule to solve the system:

4x - y + z = -5

2x + 2y + 3z = 10

5x – 2y + 6z = 1

Solution. We begin by setting up four determinants: :

D consists of the coefficients of x, y, and z from the three equations

is obtained by replacing the x-coefficients in the first column of D with the constants from the right sides of the equations.

is obtained by replacing the y-coefficients in the second column of D with the constants from the right sides of the equations

is obtained by replacing the z-coefficients in the third column of D with the constants from the right sides of the equations

Next, we evaluate the four determinants:

= 4(12 – (-6)) + 1(12 – 15) + 1(-4 – 10)

= 4(18) + 1(-3) + 1(-14)

= 72 – 3 – 14

= 55

= -5(12 – (-6)) + 1(60 – 3) + 1(-20 – 2)

= -5(18)+1(57) + 1(-22)

= -90 + 57 – 22

= -55

= 4(60 – 3) + 5(12 – 15) + 1(2 – 50)

= 4(57) + 5(-3) + 1(-48)

= 228 - 15 – 48

= 165

= 4(2 – (-20)) + 1(2 – 50) – 5(-4 – 10)

= 4(22) + 1(-48) – 5(-14)

= 88 – 48 + 70

= 110

Substitute these four values into the formula from Cramer’s Rule:

The solution is (-1, 3, 2).