Question

6x + 2y - 2z = 10

-x - 3y+7z = -27
3x + 5y - 6z = 18
Give your answer as an ordered triple.

Answer 1

Answer: (1, -3, -5)

Explanation:

Hi!

There are multiple ways to answer this question. The method we will use today is Gaussian Elimination and Row Echelon form. This requires the use of linear algebra and matrices. The main goal is to end with
`\left[\begin{array}c1&0&0&x\\0&1&0&y\\0&0&1&z\end{array}\right]`, which will give us the value of x, y, z.

`\left[\begin{array}ccc6&2&-2&10\\-1&-3&7&-27\\3&5&-6&18\end{array}\right]`

Notice that the system of three equations are written so that the first column represents x, second column represents y, third column represents z, and the fourth column represents the answer. In the following notation, R represent the different rows. Double arrow means the switching of the respective rows.

`R_1\leftrightarrow R_2\left[\begin{array}ccc-1&-3&7&-27\\6&2&-2&10\\3&5&-6&18\end{array}\right]\\`

`-R_1\left[\begin{array}c1&3&-7&27\\6&2&-2&10\\3&5&-6&18\end{array}\right] \\`

`-6R_1+ R_2 \rightarrow R_2\left[\begin{array}ccc1&3&-7&27\\0&-16&40&-152\\3&5&-6&18\end{array}\right] \\`

`-3R_1+ R_3\rightarrow R_3\left[\begin{array}ccc1&3&-7&27\\0&-16&40&-152\\0&-4&15&-63\end{array}\right]\\`

`-(1)/(16) R_2\left[\begin{array}ccc1&3&-7&27\\0&1&-(5)/(2)&(19)/(2)\\0&-4&15&-63\end{array}\right]\\`

`4R_2+R_3\rightarrow R_3\left[\begin{array}ccc1&3&-7&27\\0&1&-(5)/(2)&(19)/(2)\\0&0&5&-25\end{array}\right]\\`

`(1)/(5)R_3\left[\begin{array}c1&3&-7&27\\0&1&-(5)/(2)&(19)/(2)\\0&0&1&-5\end{array}\right]\\`

`(5)/(2)R_3+R_2\rightarrow R_2\left[\begin{array}c1&3&-7&27\\0&1&0&-3\\0&0&1&-5\end{array}\right]\\`

`7R_3+R_1\rightarrow R_1\left[\begin{array}ccc1&3&0&-8\\0&1&0&-3\\0&0&1&-5\end{array}\right]\\`

`-3R_2+R_1\rightarrow R_1\left[\begin{array}ccc1&0&0&1\\0&1&0&-3\\0&0&1&-5\end{array}\right]\\`

Keep note that our resulting matrix is the same as the very first one we started with, but we have actual values for x, y, z. Putting everything together, we get x=1, y=-3, z=-5. Since the questions asks the answer to be put into ordered triple, we have (1, -3, -5) as our final answer.

To check our answer, we just plug those values into each equation to see if they are true.

Equation 1: True

6x+2y-2z=10 [plug in x, y, z]

6(1)+2(-3)-2(-5)=10 [multiply]

6-6+10=10 [subtract]

0+10=10 [add]

10=10

------------------------------------------------------------

Equation 2: True

-x-3y+7z=-27 [plug in x, y, z]

-(1)-3(-3)+7(-5)=-27 [multiply]

-1+9-35=-27 [add]

8-35=-27 [subtract]

-27=-27

------------------------------------------------------------

Equation 3: True

3x+5y-6z=18 [plug in x, y, z]

3(1)+5(-3)-6(-5)=18 [multiply]

3-15+30=18 [subtract]

-12+30=18 [add]

18=18

Checking all our answers, every single equation was true. Therefore, our final answer of (1, -3, -5) is correct.