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.

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asked Feb 26, 2023 7.9k views

1 Answer

Mtoossi · Answer 1 · 2023-03-04T14:52:13+0000

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

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:

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