# Thread: [SOLVED] using matrices to solve for a system of equations using gauss-jordan elimina

1. ## [SOLVED] using matrices to solve for a system of equations using gauss-jordan elimina

i have to use matrices to solve the system of equations using gauss-jordan elimination

my system of equations is:
2x -y +3z =24
2y -z =14
7x -5y =6

my matrix would be
2 -1 3 24
0 2 -1 14
7 -5 0 6

can i have a walk through of this equations and an explanation on what to do? this is what i have so far:

i want to get this matrix into reduced row-echelon form, so first i put it into row-echelon form then reduce.
-i times Row 1 by 1/2 so i would end up with:
1 -1/2 3/2 12
0 2 -1 14
7 -5 0 6

-then i times Row 2 by 1/2 and end up with:
1 -1/2 3/2 12
0 1 -1/2 7
7 -5 0 6

-then i times Row 1 by -7 and add it to Row 3 and end up with:
1 -1/2 3/2 12
0 2 -1 14
0 -1.5 -10.5 -78

what would i do to get the leading 1 of row 3? am i doing this right? are my answers right so far? can anyone give me a walkthrough through the rest of the problem? sorry, my spacing isn't working.

2. calculator answer=........ x = 8, y = 10, and z = 6

3. Originally Posted by somanyquestions
i have to use matrices to solve the system of equations using gauss-jordan elimination

my system of equations is:
2x -y +3z =24
2y -z =14
7x -5y =6

my matrix would be
2 -1 3 24
0 2 -1 14
7 -5 0 6

can i have a walk through of this equations and an explanation on what to do? this is what i have so far:

i want to get this matrix into reduced row-echelon form, so first i put it into row-echelon form then reduce.
-i times Row 1 by 1/2 so i would end up with:
1 -1/2 3/2 12
0 2 -1 14
7 -5 0 6

-then i times Row 2 by 1/2 and end up with:
1 -1/2 3/2 12
0 1 -1/2 7
7 -5 0 6

-then i times Row 1 by -7 and add it to Row 3 and end up with:
1 -1/2 3/2 12
0 2 -1 14
0 -1.5 -10.5 -78

what would i do to get the leading 1 of row 3? am i doing this right? are my answers right so far? can anyone give me a walkthrough through the rest of the problem? sorry, my spacing isn't working.
$\left[\begin{matrix}
2 & -1 & 3 &|& 24\\
0 & 2 & -1 &|& 14\\
7 & -5 & 0 &|& 6 \end{matrix}\right]$

$\frac{1}{2}R_1 \to R_1$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 2 & -1 &|& 14\\
7 & -5 & 0 &|& 6 \end{matrix}\right]$

$R_3 - 7R_1 \to R_3$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 2 & -1 &|& 14\\
0 & -\frac{3}{2} & -\frac{21}{2} &|& -78 \end{matrix}\right]$

$\frac{1}{2}R_2 \to R_2$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 1 & -\frac{1}{2} &|& 7\\
0 & -\frac{3}{2} & -\frac{21}{2} &|& -78 \end{matrix}\right]$

$R_3 + \frac{3}{2}R_2 \to R_3$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 1 & -\frac{1}{2} &|& 7\\
0 & 0 & -\frac{45}{4} &|& -\frac{135}{2} \end{matrix}\right]$

$-\frac{4}{45}R_3 \to R_3$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 1 & -\frac{1}{2} &|& 7\\
0 & 0 & 1 &|& 6 \end{matrix}\right]$

Now it is in row-echelon form and you can solve the system of equations.

4. Originally Posted by Prove It
$\left[\begin{matrix}
2 & -1 & 3 &|& 24\\
0 & 2 & -1 &|& 14\\
7 & -5 & 0 &|& 6 \end{matrix}\right]$

$\frac{1}{2}R_1 \to R_1$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 2 & -1 &|& 14\\
7 & -5 & 0 &|& 6 \end{matrix}\right]$

$R_3 - 7R_1 \to R_3$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 2 & -1 &|& 14\\
0 & -\frac{3}{2} & -\frac{21}{2} &|& -78 \end{matrix}\right]$

$\frac{1}{2}R_2 \to R_2$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 1 & -\frac{1}{2} &|& 7\\
0 & -\frac{3}{2} & -\frac{21}{2} &|& -78 \end{matrix}\right]$

$R_3 + \frac{3}{2}R_2 \to R_3$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 1 & -\frac{1}{2} &|& 7\\
0 & 0 & -\frac{45}{4} &|& -\frac{135}{2} \end{matrix}\right]$

$-\frac{4}{45}R_3 \to R_3$

$\left[\begin{matrix}
1 & -\frac{1}{2} & \frac{3}{2} &|& 12\\
0 & 1 & -\frac{1}{2} &|& 7\\
0 & 0 & 1 &|& 6 \end{matrix}\right]$

Now it is in row-echelon form and you can solve the system of equations.
how do you type out all these matrices?

5. Originally Posted by somanyquestions
how do you type out all these matrices?
Click on them - that gives you the code I used.

7. Originally Posted by somanyquestions