Solve the system of linear equations using the gauss-jordan elimination method.
2x + 2y + z = 18
x + z = 7
4y − 3z = 20

Values for each variable are:

x = 19

y = -4

z = -12

Step-by-step explanation:

As we can remember the Gauss-Jordan elimination method consists of creating a matrix with all the equations of the system.  Remember that, if a variable does not appear in one of the equations, we give a value of 0 to its coefficient .  Each equation will constitute a line of the matrix. So, the matrix will look like this:

2   2   1   18

1    0   1    7

0   4  -3  20

For the Gauss-Jordan elimination we can multiply lines, add or subtract one line to another or we can rearrange the order at any given time. The goal is to get only 1s in the matrix diagonal, to determine the value of each variable.

Since we already have a line with a 1, we'll take that line as our starting point, and we'll rearrange it as our 1st line. By multiplying the 1st line for  2 and then subtracting the result to the second line:

1   0   1   7

0   2  -1  4

0   4  -3  20

Now, we multiply the second line by 2 and subtract the result to the third line

1   0   1   7

0   2  -1  4

0   0  -1  12

In order to get the value of Z all we have to do is multiply the third line by (-1).

1   0   1   7

0   2  -1  4

0   0  1  -12

Now, we add the third line to the second line.

1   0   1   7

0   2  0  -8

0   0  1  -12

Then, multiply the second line by a fraction 1/2, to get the value for Y

1   0   1   7

0   1  0  -4

0   0  1  -12

Finally, we subtract the third line to the 1st line to get the value for X

1   0   0  19

0   1  0  -4

0   0  1  -12

All we got left is to prove our answer is correct by replacing the variables in the system with the values found:

First equation

2(19) + 2(-4) + (-12) = 18

38 - 8 - 12 = 18

38 - 20 = 18

Second equation

19 + (-12) = 7

19 -12 = 7

Third equation

4(-4) - 3(-12) = 20

-16 + 36 = 20

(x, y)→(x − 8, y − 7)

step-by-step explanation:

enjoy!