Question

Solve the system. given your answer as (x, y, z)-4x -y - 3z = -5-6x + y - 3z = -172x + 2y - z = - 10

Answer 1

(1, -5 ,2)

Step-by-step explanation:

Given the system of equations:

`\begin{gathered} -4x-y-3z=-5\ldots(1) \\ -6x+y-3z=-17\ldots(2) \\ 2x+2y-z=-10\ldots(3) \end{gathered}`

Make z the subject in the third equation:

`z=2x+2y+10`

Substitute z=2x+2y+10 into the first and second equations:

First Equation

`\begin{gathered} -4x-y-3z=-5 \\ -4x-y-3(2x+2y+10)=-5 \\ -4x-y-6x-6y-30=-5 \\ -4x-6x-y-6y=-5+30 \\ -10x-7y=25\ldots(4) \end{gathered}`

Second Equation

`\begin{gathered} -6x+y-3z=-17 \\ -6x+y-3(2x+2y+10)=-17 \\ -6x+y-6x-6y-30=-17 \\ -6x-6x+y-6y=-17+30 \\ -12x-5y=13\ldots(5) \end{gathered}`

Next, solve equations 4 and 5 simultaneously:

`\begin{gathered} -10x-7y=25\ldots(4) \\ -12x-5y=13\ldots(5) \end{gathered}`

Multiply equation (4) by 5 and equation (5) by 7.

`\begin{gathered} -50x-35y=125 \\ -84x-35y=91 \\ \text{Subtract same sign} \\ 34x=34 \\ x=(34)/(34) \\ x=1 \end{gathered}`

Substitute x=1 into equation (4):

`\begin{gathered} -10x-7y=25\ldots(4) \\ -10(1)-7y=25 \\ -7y=25+10 \\ -7y=35 \\ y=(35)/(-7) \\ y=-5 \end{gathered}`

Recall: z=2x+2y+10

`\begin{gathered} z=2x+2y+10 \\ =2(1)+2(-5)+10 \\ =2-10+10 \\ z=2 \end{gathered}`

The solution of the system is:

`(1,-5,2)`