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,
$x+y+z=3$
$x+2y+3z=0$
$x+3y+2z=3$
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,
$2y+4z=6$→$2y+4(2)=6$
→$2y8=6$
→$2y=2$
→$y=1$
So now we know that $z=2$, and $y=1$. Now let us take a look at the first row and solve for $x$.
$x+y+z=3$→$x+12=3$
→$x1=3$
→$x=4$
Since we have solved for $x,y$ and $z$, then we have just solved the linear system.

2.
Gaussian Elimination
Solve the following linear systems: