- 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, 4) satifies all 3 equations.
- Plug x = 1, y = 1, and z = 4 into equations (1), (2), and (3), and all
equations hold true.
- (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.
- 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:
- Solve any one of the equations for any one of the variables.
- Substitute into the other two equations: the expression for the
variable found in step 1.
- Solve the resulting 2 equations in 2 variables by the methods of
section 4.1.
- 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.
- You now have values for the 3 variables (i.e., you have the solution).
Check that the solution satisfies all 3 equations.
- Another way:
- 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.
- 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.
- Now you have 2 equations in 2 variables, so solve these by the methods of
section 4.1.
- 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.
- You now have values for the 3 variables (i.e., you have the solution).
Check that the solution satisfies all 3 equations.
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