Math 101 Intermediate Algebra    Third-Order Systems of Linear Equations
Chapter 4, Section 2

General Ideas

"Third-Order Systems of Linear Equations" is just a fancy way of saying
• 3 equations in 3 unknowns, or
• 3 equations in 3 variables.

An example of a system of 3 equations in 3 unknowns:

 3x + 7y - z = 6 (1) 2x - 3y + 2z = 7 (2) -2x - 2y + 3z = 8 (3)

The 3 unknows are x, y, and z. A solution will be an ordered triple (x, y, z).

Observe that for the above example:

1. (1, 1, 4) satifies all 3 equations.
• Plug x = 1, y = 1, and z = 4 into equations (1), (2), and (3), and all equations hold true.
2. (0, 0, 1) does not satisfy any of the 3 equations, so it can't be the solution.
• Plug x = 0, y = 0, and z = 1 into equations (1), (2), and (3), and none of the equations are true.
3. Conclusion:
• (1, 1, 4) is a solution to the example system of equations.
• (0, 0, 1) is not a solution to the example system of equations.

How to Solve 3 Equations in 3 Unknowns

One way:

1. Solve any one of the equations for any one of the variables.
2. Substitute into the other two equations:  the expression for the variable found in step 1.
3. Solve the resulting 2 equations in 2 variables by the methods of section 4.1.
4. Step 3 obtains values for 2 of the variables. Plug these values into one of the original equations to get the value of the third variable.
5. You now have values for the 3 variables (i.e., you have the solution). Check that the solution satisfies all 3 equations.

Another way:

1. Eliminate a variable from any 2 of the 3 equations (by the elimination method of section 4.1). You choose which 2 of the 3 equations to use.
2. Eliminate the same variable from the remaining equation (the equation not used in step 1) and either of the 2 equations used in step 1. Again, use the elimination method of section 4.1.
3. Now you have 2 equations in 2 variables, so solve these by the methods of section 4.1.
4. Step 3 obtains values for 2 of the variables. Plug these values into one of the original equations to get the value of the third variable.
5. You now have values for the 3 variables (i.e., you have the solution). Check that the solution satisfies all 3 equations.