1. Systems of Linear Equations The solution will be one of three cases: 1. Exactly one solution, an ordered triple (x, y, z) 2. A dependent system with infinitely many solutions 3. No solution Three Equations Containing Three Variables As before, the first two cases are called consistent since there are solutions. The last case is called inconsistent.
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.
3. Planes intersect in a line: consistent system called dependent with an infinite number of solutions
6. We will be doing elimination to solve these systems. It’s like elimination that you learned with two equations and variables but it’s now the problem is bigger. 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 . coefficient is a 1 here so it will be easy to work with
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. 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. 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.
9. 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 keep over here for later use We can add this to the one’s we’ve kept up in the corner
10. 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 . keep over here for later use 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. Let’s do another one: we’ll keep this one Now we’ll use the 2 equations we have with y and z to eliminate the y ’s. If we multiply the equation we kept by 2 and add it to the first equation we can eliminate x ’s. 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.
12. 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. 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.
13. Let’s do another one: we’ll keep this one If we multiply the equation we kept by -2 and add it to the second equation we can eliminate x ’s. Now we’ll use the 2 equations we have with y and z to eliminate the y ’s. If we multiply the equation we kept by -4 and add it to the last equation we can eliminate x ’s. let's “keep” this one since it will be easy to use to eliminate x ’s from others.
14. 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. 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 .
15. First we just put z = z since it can be any real number. Now solve for y in terms of z . Now sub it - z for y in first equation and solve for x in terms of z . The solution is (1 - z , - z , z ) where z 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.
