1. Solving Systems of Linear equations with 3 Variables To solve for three variables, we need a system of three independent equations.

2. Independent, Dependent and Inconsistent Equations Systems of three equations in three variables follow the same characteristics of systems of equations in two variables – Independent equations have one solution – Dependent equations have an infinite number of solutions – Inconsistent equations have no solution Solving a system of equations in three variables involves a few more steps, but is essentially the same process as for systems of two equations in two variables.

3. Steps to solving a system of equations in 3 Variables by Elimination Ensure that the equations are in standard form: Ax + By + Cz = D Eliminate one of the variables using two of the three equations. Result will be a new equation with two variables. Eliminate the same variable using another set of two equations. Result will be a second equation in two variables. Solve the new system of two equations. Using the solution for the two variables, substitute the values into one of the original equations to solve for the third variable. Check the solution set in the remaining two original equations.

4. Example system of three equations First, we need to ensure that all equations are in standard form, i.e. all variables are on the left side of the equation. Note that equation #3 is not in standard form. We need to get the y and z terms to the left side of the equation. Use elimination to solve the following system of equations: x + 2y - 3z = -2 2x - 2y + z = 7 x + y + 2z = -4

5. Use elimination to solve the following system of equations: -x + y + 2z = 7 2x + 3y + z = 1 -3x - 4y + z = 4

6. Use elimination to solve the following system of equations: -x + y + 2z = 7 2x + 3y + z = 1 -3x - 4y + z = 4 Solution (-2, 1, 2)

7. Steps to solving a system of equations in 3 Variables by Cramer’s Rule To apply Cramer’s rule to 3 x 3 systems, you need to find the determinant of a 3 x 3 matrix. To find the determinant: 1 st step – Rewrite the first two columns at the right side of the determinant. 2 nd step - Add the products of each down diagonal. 3 rd step – Subtract the sum of the up diagonal. Cramer’s Rule can be expanded to cover 3 x 3 systems.

8. Use Cramer’s Rule to solve the following system of equations: 4x - 2y + z = -6 3x + 3y + 5z = -8 2x - 4y - 3z = 2

9. Use Cramer’s Rule to solve the following system of equations: 4x - 2y + z = -6 3x + 3y + 5z = -8 2x - 4y - 3z = 2 Solution (1, 3, -4)