Solving a linear system with matrices using Gaussian elimination - Matrices

Solving a linear system with matrices using Gaussian elimination

Now that we have learned how to represent a linear system as a matrix, we can now solve this matrix to solve the linear system! We use a method called "Gaussian elimination". This method involves a lot of matrix row operations. Our goal is to make it so that all entries in the bottom left of the matrix are 0. Once that is done, we take a look at the last row and convert it to a linear system. Then we solve for the variable. Then we look at the second last row, convert it to a linear system, and solve for the other variable. Rinse and repeat, and you will find all the variables which solve the linear system!

Lessons

Notes:
Note
Gaussian elimination (or row reduction) is a method used for solving linear systems. For example,

x+y+z=3x+y+z=3
x+2y+3z=0x+2y+3z=0
x+3y+2z=3x+3y+2z=3

Can be represented as the matrix:
linear system in matrix form

Using Gaussian elimination, we can turn this matrix into

applying gaussian elimination to a matrix (watch the intro video to learn how to do this!)

Now we can start solving for x,yx,y and zz.

So in the third row, we see that 3z=6-3z=6. So z=2z=-2.

In the second row, we see that 2y+4z=62y+4z=-6. Since we know that z=2z=-2, then we can substitute it into the second row and solve for yy. So,

2y+4z=62y+4z=-6 2y+4(2)=6 2y+4(-2)=-6
2y8=6 2y-8=-6
2y=2 2y=2
y=1 y=1

So now we know that z=2z=-2, and y=1y=1. Now let us take a look at the first row and solve for xx.

x+y+z=3x+y+z=3 x+12=3 x+1-2=3
x1=3 x-1=3
x=4 x=4

Since we have solved for x,yx,y and zz, then we have just solved the linear system.
  • 2.
    Gaussian Elimination
    Solve the following linear systems:
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Solving a linear system with matrices using Gaussian elimination

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