The document provides examples of solving systems of linear equations in three variables through algebraic manipulation steps like addition, subtraction, multiplication, and division. The key steps are to eliminate variables between equations until reaching a solution or determining there is no solution/infinite solutions. Example solutions provided are (–2, 6, –3), (–1, 2, –4), and determinations of no solution or infinite solutions based on whether the resulting equations are or aren't identities.

1. First equationThird equationSubtract to eliminate z.Notice that the z terms in each equation have been eliminated. The result istwo equations with the two same variables x and y.Example 5-1a

2. Add to eliminate y.Divide by 29.Equation with two variablesMultiply by 5.Replace x with –2.Multiply.Simplify.Step 2Solve the system of two equations. Substitute –2 for x in one of the two equations with two variables and solve for y.Example 5-1a

3. Equation with three variablesReplace x with –2 and y with 6.Multiply.Simplify.Step 3Substitute –2 for x and 6 for yin one of the original equations with three variables.Answer: The solution is (–2, 6, –3). You can check this solution in the other two original equations.Example 5-1a

4. Solve the system of equations.Pick a variable to eliminate from all three equations.Example 5-1b

5. Solve the system of equations.Use two equations at a time to eliminate the same variable.2x + 3y – 3z = 16−2x – 2y – 2z = 62x + 3y – 3z = 16 x + y + z = – 3• – 2y – 5z = 22Example 5-1b

6. Solve the system of equations.Use two other equations to eliminate the same variable.2x + 3y – 3z = 16−2x + 4y + 2z = 22x + 3y – 3z = 16 x – 2y – z = – 1• – 27y – z = 18Example 5-1b

7. Solve the new system of equations.y – 5z = 22y – 5z = 22• – 57y – z = 18– 35y+5z = – 90– 34y = – 68 y = 2 7(2) – z = 1814 – z = 18– z = 4z = – 4

8. y = 2 z = – 4Solve for the remaining variable.x + y + z = – 3x + 2 + – 4 = – 3x – 2 = – 3x = – 1Answer:(–1, 2, –4)Example 5-1b

9. Answer:(–1, 2, –4)Check!!!2(– 1) + 3(2) – 3(–4) = 16– 1 + 2 + –4 = – 3– 1 – 2(2) – (–4) = – 1Example 5-1b

10. Solve the system of equations.Answer: The solution is (–2, 6, –3)..Example 5-1a

11. Solve the system of equations.Multiply by 3.Multiply by 2.Answer: The equation is never true. So, there is no solution of this system.Eliminate x in the second two equations.Example 5-3a

12. Solve the system of equations.Multiply by 3.Eliminate y in the first and third equations.Example 5-2a

13. The equation is always true. This indicates that the first and third equations represent the same plane. Check to see if this plane intersects the second plane.Divide by the GCF, 3.Multiply by 6.Answer: The planes intersect in a line. So, there are an infinite number of solutions.Example 5-2a

14. Solve the system of equations.Answer:There are an infinite number of solutions.Example 5-2b

15. Solve the system of equations.Answer:There is no solution of this system.Example 5-3b