Question

Solve the system of linear equations using the Gauss-Jordan elimination method.

(x,y,z)=__________________

2x + 2y − 3z = 16
2x − 3y + 2z = −4
4x − y + 3z =
−4

Answer 1

(x,y,z)=(2,0,-4)

Explanation:

• First we create the extended matrix from the equations

`\left[\begin{array}ccc2&2&-3&16\\2&-3&2&-4\\4&-1&3&-4\end{array}\right]`

Using the elementary operations

• Substract to the 2nd line the first one, and the 3rd one twice the first:

`\left[\begin{array}c2&2&-3&16\\0&-5&5&-20\\0&-5&9&-36\end{array}\right]`

• Divide the first line by 2, the 2nd one by -5 and substract to the 3rd the 2nd:

`\left[\begin{array}ccc1&1&-3/2&8\\0&1&-1&4\\0&0&4&-16\\\end{array}\right]`

• Divide the 3rd by 4:

`\left[\begin{array}ccc1&1&-3/2&8\\0&1&-1&4\\0&0&1&-4\\\end{array}\right]`

• Add the 3rd to the 2nd:

`\left[\begin{array}ccc1&1&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right]`

• Substract the 2nd to the 1st

`\left[\begin{array}c1&0&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right]`

• Add the 3rd multiplied by 3/2:

`\left[\begin{array}c1&0&0&2\\0&1&0&0\\0&0&1&-4\\\end{array}\right]`

The answer is determined:

x=2

y=0

z=-4

You can check they are correct, by entering in the original formulas.