Solving a linear system with matrices using Gaussian elimination

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

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

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

Can be represented as the matrix:

Using Gaussian elimination, we can turn this matrix into

(watch the intro video to learn how to do this!)

Now we can start solving for $x,y$ and $z$.

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

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

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

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

$x+y+z=3$$x+1-2=3$
$x-1=3$
$x=4$

Since we have solved for $x,y$ and $z$, then we have just solved the linear system.
• Introduction
Gaussian elimination overview

• 1.
Gaussian Elimination
Solve the following linear systems:
a)
$x+2y=3$
$2x+3y=1$

b)
$x+4y+3z=1$
$x+2y+9z=1$
$x+6y+6z=1$

c)
$x+3y+3z=2$
$3x+9y+3z=3$
$3x+6y+6z=4$

d)
$4x-5y=-6$
$2x-2y=1$

e)
$x+3y+4z=4$
$-x+3y+2z=2$
$3x+9y+6z=-6$