Question

I need to learn how to solve a system of equations using substitution. Here are the equations.-1x -1y -1z = -8-4x + 4y +5z = 72x + 0y +2z = 4

Answer 1

x = 3, y = 6, and z = *1

Step-by-step explanation:

The substitution methods ask that we solve for one or two variables in one equation and substitute its value in the second equation.

For example, take the set of equations we are given to solve for x, y, and z:

`\begin{gathered} -x-y-z=8 \\ -4x+4y+5z=7 \\ 2x+0y+2z=4 \end{gathered}`

Now we take a look at the third equation.

`2x+0+2z=4`
`\Rightarrow2x+2z=4`

dividing both sides by 2 gives

`x+z=2`

further multiplying both sides by -1 gives

`-x-z=-2`

Now we substitute this value into -1x -1y -1z = -8.

`-x-y-z=-8\Rightarrow-x-z-y=-8`

substitution gives

`-2-y=-8`

Now we can solve for y:

adding 2 to both sides gives

`-y=-8+2`
`-y=-6`

which ( upon multiplying both sides by -1 ) gives

`\boxed{y=6.}`

Now we know the value of y. How do we find the value of x and z?

Putting in the value of y into the first and the second equation gives

`\begin{gathered} -1x-1(6)-1z=-8 \\ -4x+4(6)+5z=7 \end{gathered}`

which gives us

`\begin{gathered} -x-6-z=-8\Rightarrow-x-z=-2 \\ -4x+24+5z=7\Rightarrow-4x+5z=-17 \end{gathered}`

Now our problem has been reduced to two equations and two unknowns.

Solving for x in the first equation above gives

`x=z-2`

Substituting that into the second equation gives

`-4(z-2))+5z=-17`
`4(z-2)+5z=-17`

solving fro z gives

`z=-1`

with the value of z in hand, we put it into the x = z-2) to get

`x=(2--1)`
`\Rightarrow3`

Hence, the solution to the system is

x = 3, y = 6, and z = -1