Question

Write a system of three linear equations in three variables that has the ordered triple (-4, 1, 2) as its only solution. Justify your answer using the substitution method.

Answer 1

We have the condition that the only solution for the SEL is the ordered triple:

`(x,y,z)=(-4,1,2)`

One simple example of a system of equations that has this solution can be:

`\begin{gathered} x+y=-3 \\ x+z=-2 \\ y+z=3 \end{gathered}`

This can be written as:

`\begin{gathered} 1\cdot x+1\cdot y+0\cdot z=-3 \\ 1\cdot x+0\cdot y+1\cdot z=-2 \\ 0\cdot x+1\cdot y+1\cdot z=3 \end{gathered}`

Applying the substitution method, we start by substituting x in the first two equations:

`\begin{gathered} x+y=-3\longrightarrow y=-3-x \\ x+z=-2\longrightarrow z=-2-x \end{gathered}`

Then, as we used the information of the first two equations, we replace in the third equation:

`\begin{gathered} y+z=(-3-x)+(-2-x)=3 \\ -5-2x=3 \\ -2x=3+5 \\ -2x=8 \\ x=(8)/(-2) \\ x=-4 \end{gathered}`

As know we know the value of x, we can replace in the first two equations:

`x+y=-3\longrightarrow y=-3-x=-3-(-4)=-3+4=1`
`x+z=-2\longrightarrow z=-2-x=-2-(-4)=-2+4=2`

Then we have the solution we where looking for:

`\begin{gathered} x=-4 \\ y=1 \\ z=2 \end{gathered}`