Question

Solve the following system of linear equations.-x + 2y + z = 122x – 2y - 3z = - 183x - 5y + 4z = 20AnswerKeypaKeyboard ShortX =y =z =

Answer 1

We need to solve the next system of linear equations:

`\begin{gathered} x+2y+z=12 \\ 2x-2y-3z=-18 \\ 3x-5y+4z=20 \end{gathered}`

Let's pair the equations to eliminate 1 variable.

The first pair, add both equations:

`\begin{gathered} x+2y+z=12 \\ 2x-2y-3z=-18 \\ --------------- \\ (x+2x)+(2y-2y)+(z-3z)=12-18 \\ -------------------------- \\ 3x+0-2z=-6 \end{gathered}`

The second pair:

`\begin{gathered} 2x-2x-3z=-18 \\ 3x-5y+4z=20 \end{gathered}`

Multiply the first equation by 5 and the second equation by -2:

`\begin{gathered} 5(2x-2x-3z=-18) \\ -2(3x-5y+4z=20) \end{gathered}`

Then, add both equations:

`\begin{gathered} 10x-10y-15z=-90 \\ -6x+10y-8z=-40 \\ -------------- \\ (10x-6x)+(-10y+10y)+(-15z-8z)=(-90-40) \\ ---------------------------- \\ 4x+0-23z=-130 \end{gathered}`

Now, solve the new system :

`\begin{gathered} 3x+0-2z=-6 \\ 4x+0-23z=-130 \end{gathered}`

Multiply the first equation by 4 and multiply the second equation by -3:

`\begin{gathered} 4(3x+0-2z=-6) \\ -3(4x+0-23z=-130) \end{gathered}`
`\begin{gathered} 12x-8z=-24 \\ -12x+69z=390 \end{gathered}`

Add both equations:

`\begin{gathered} (12x-12x)+(-8z+69z)=(-24+390) \\ -------------------------- \\ 0+61z=366 \end{gathered}`

Solve for z:

`\begin{gathered} z=(366)/(61) \\ \text{Then} \\ z=6 \end{gathered}`

To solve for x, replace the z value on one equation:

`\begin{gathered} 4x+0-23(6)=-130 \\ \text{Then} \\ 4x-138=-130 \\ 4=-130+138 \\ 4x=8 \\ x=(8)/(4) \\ x=2 \end{gathered}`

Finally, replace the z and x values:

`\begin{gathered} x+2y+z=12 \\ 2+2y+6=12 \\ \end{gathered}`

Solve for y:

`\begin{gathered} 8+2y=12 \\ 2y=12-8 \\ 2y=4 \\ y=(4)/(2) \\ y=2 \end{gathered}`

Hence, the result for each variable are:

x= 2

y= 2

z=6