Transcript 3.6 Solving Systems of Linear Equations in 3 Variables
Solving Systems of Linear Equations in 3 Variables
Pictured below is an example of the graph of the plane 2x + 3y + z = 6. The red triangle is the portion of the plane when x, y, and z values are all positive. This plane actually continues off in the negative direction. All that is pictured is the part of the plane that is intersected by the positive axes (the negative axes have dashed lines).
Like systems of linear equations, the solution of a system of planes can be no solution, one solution or infinite solutions. No Solution of three variable systems There is no single point at which
all three
planes intersect, therefore this system has no solution.
One Solution of three variable systems If three planes intersect, then the three variable system has 1 point in common, and a single solution represented by the black point below.
Infinite Solutions of three variable systems If the three planes intersect, then the three variable system has a line of intersection and therefore an infinite number of solutions.
A system of linear equations in 3 variables looks something like: x + 3y – z = -11 2x + y + z = 1 5x – 2y + 3z = 21
A solution is an ordered triple (x, y, z) that makes all 3 equations true.
Is (1, 1, 1) a solution?
x
2
x z
3 4
z
5
x
4
y
2
z
3
Yes
Is (2, 1, 6) a solution?
x
2
x y z
3
z
9 4
x z
15
No
Steps for solving in 3 variables 1. Using the 1 st 2 equations, cancel one of the variables.
2. Using the last 2 equations, cancel the same variable from step 1.
3. Use the results of steps 1 & 2 to solve for the 2 remaining variables.
4. Plug the results from step 3 into one of the original 3 equations and solve for the 3 rd remaining variable.
5. Write the solution as an ordered triple (x,y,z).
1. Solve the system.
x
3
y z
11 2
x z
5
x
2
y
3
z
1 21
(2, -4, 1)
2. Solve the system.
x
2
x
3
x z
5
z
2
z
4 8
(1, 1, 3)
3. Solve the system.
2
y
2
x
2
y
4
x
4
y
2
z
3 5 6
No solution
4. Solve the system.
2
x
2
x x
3
y z
7
y y
3
z
2
z
17 12
(-1, 0, 5)
5. Solve the system.
x x
3
z
2 1
z
2
y z
15 4
(-2, -8, -1)