Answer to Question #326077 in Linear Algebra for Arrdly

Question

Determine which of the following is the solution set of the linear equations below.

3x − y + z = 2

2x − z = 2

Answer 1

The system has an infinite number of solutions because we have 3 variables and only 2 equations:

$\begin{cases}3x-y+z=2\\ 2x-z=2 \end{cases} \implies \begin{cases} 5x-y=4\\ 2x-z=2 \end{cases} \implies$$\begin{cases} 5x-y=4\\ x=\frac12(2+z) \end{cases} \implies\begin{cases} 5x-y=4\\ x=1+0.5z \end{cases}\implies$ $\begin{cases} 5(1+0.5z)-y=4\\ x=1+0.5z \end{cases}\implies\begin{cases} 5+2.5z-y=4\\ x=1+0.5z \end{cases}$$\implies\begin{cases} y=1+2.5z\\ x=1+0.5z \end{cases}\implies$ $\begin{cases} x=1+0.5z\\ y=1+2.5z\\ z\in R \end{cases}$

$R$ - set of real numbers.