1. Solving Systems of Linear Equations in Three Variables

2. Warm-Up
No Solution
Infinitely many solutions

3. Here is a system of three linear equations in three variables:
The ordered triple (2,-1,1) is a solution to this system since it is a solution to all three equations.

4. The graph of a linear equation in three variables is a plane
The graph of a linear equation in three variables is a plane. Three planes in space can intersect in different ways
The planes could intersect in a line. The system has infinitely many solutions
The planes could intersect in a single point. The system has exactly one solution
The planes could have NO point of intersection. The left figure shows planes that intersect pairwise, but all 3 do not have a common point of intersection. The right figure shows parallel planes. Each system has NO solution.

6. Substitution Method - Example 1
Since x+y=z, substitute this for z in the first two equations
Simplify
Finally, solve this linear system of two equations and two variables to get x = 4 and y =8
Since z=x+y, z = 12. Our final solution is (4,8,12)

7. Example 2 -3(2z+10) + y – 4z = 20 -4(2z+10) + 2y + 3z = -15
Solving for x in the first equation, we get 2z + 10 =x. Substitute this for x in the last two equations
-3(2z+10) + y – 4z = 20
-4(2z+10) + 2y + 3z = -15
Simplify
Finally, solve this linear system of two equations and two variables to get y= 0 and y = -5
y – 10z = 50
2y -5z = 25
Since x=2z + 10, x = 0. Our final solution is (0, 0, -5)

8. Example 3 5r +4(t+2) – 6t = -24 -2(t+2) + 2t = 0 -2t - 4 + 2t = 0
Since t + 2 = s, substitute this for s in the first two equations
5r +4(t+2) – 6t = -24
-2(t+2) + 2t = 0
Simplify
Since we have a false statement, this system is inconsistent.
-2t t = 0
-4 = 0
Our final solution is No Solution

9. Try it on your own! B. A.