1. Systems of Linear Equations
Three Equations Containing Three Variables
The first two cases are called consistent since there are solutions. The last case is called inconsistent.
The solution will be one of three cases:
1. Exactly one solution, a point (x, y, z) 2. A dependent system with infinitely many solutions 3. No solution

2. Planes intersect at a point: consistent with one solution
With two equations and two variables, the graphs were lines and the solution (if there was one) was where the lines intersected. Graphs of three variable equations are planes. Let’s look at different possibilities. Remember the solution would be where all three planes all intersect.
Planes intersect at a point: consistent with one solution

3. Planes intersect in a line: consistent system called dependent with an infinite number of solutions

4. Three parallel planes: no intersection so system called inconsistent with no solution

5. No common intersection of all three planes: inconsistent with no solution

6. We will be doing a “monster” elimination
We will be doing a “monster” elimination. (This just means it’s like elimination that you learned with two equations and variables but it’s now “monster” because the problems are bigger and meaner and uglier).
coefficient is a 1 here so will easy to work with
Your first strategy would be to choose one equation to keep that has all 3 variables, but then use that equation to “eliminate” a variable out of the other two. I’m going to choose the last equation to “keep” because it has just x.

7. keep over here for later use
Now use the third equation multiplied through by whatever it takes to eliminate the x term from the first equation and add these two equations together. In this case when added to eliminate the x’s you’d need a −2.
put this equation up with the other one we kept

8. Now use the third equation multiplied through by whatever it takes to eliminate the x term from the second equation and add these two equations together. In this case when added to eliminate the x’s you’d need a 3.
we won’t “keep” this equation, but we’ll use it together with the one we “kept” with y and z in it to eliminate the y’s.

9. keep over here for later use
So we’ll now eliminate y’s from the 2 equations in y and z that we’ve obtained by multiplying the first by 4 and the second by 5
we can add this to the one’s we’ve kept up in the corner

10. keep over here for later use
Now we are ready to take the equations in the corner and “back substitute” using the equation at the bottom and substituting it into the equation above to find y.
Now we know both y and z we can sub them in the first equation and find x
These planes intersect at a point, namely the point (2 , −1 , 1). The equations then have this unique solution. This is the ONLY x, y and z that make all 3 equations true.

11. The planes intersect in a point.

12. Let’s do another one:
I’m going to “keep” this one since it will be easy to use to eliminate x’s from others.
If we multiply the equation we kept by 3 and add it to the last equation we can eliminate x’s.
If we multiply the equation we kept by 2 and add it to the first equation we can eliminate x’s.
we’ll keep this one
Now we’ll use the 2 equations we have with y and z to eliminate the y’s.

13. The system is inconsistent,
no common intersection between the three planes.

14. This means the equations are inconsistent and have no solution
This means the equations are inconsistent and have no solution. The planes don’t have a common intersection and there is not any (x, y, z) that make all 3 equations true.
we’ll multiply the first equation by -2 and add these together
Oops---we eliminated the y’s alright but the z’s ended up being eliminated too and we got a false equation.

15. Let’s do another one:
I’m going to “keep” this one since it will be easy to use to eliminate x’s from others.
If we multiply the equation we kept by −4 and add it to the last equation we can eliminate x’s.
If we multiply the equation we kept by −2 and add it to the second equation we can eliminate x’s.
we’ll keep this one
Now we’ll use the 2 equations we have with y and z to eliminate the y’s.

16. This means the equations are consistent and have infinitely many solutions. The planes intersect in a line. To find points on the line we can solve the 2 equations we saved for x and y in terms of z.
multiply the first equation by −1 and add the equations to eliminate y.
Oops---we eliminated the y’s alright but the z’s ended up being eliminated too but this time we got a true equation.

17. First we just put z = t since it can be any real number
First we just put z = t since it can be any real number. Now solve for y in terms of z.
Now sub it −t for y in first equation and solve for x in terms of t.
The solution is (1 − t , −t , t) where t is any real number.
For example: Let z be 1. Then (0 , −1 , 1) would be a solution. Notice is works in all 3 equations.
But so would the point you get when z = 2 or 3 or any other real number so there are infinitely many solutions.

18. The three planes intersect in the line.