Question:In Problems 17 to 20, solve the sets of homogeneous equations by row reducing the matrix.

18.$\{\begin{array}{c}\text{2}x+\text{3}z=\text{0}\\ \text{4}x+\text{2}y+\text{5}z=\text{0}\\ x-y+\text{2}z=\text{0}\end{array}$

A linear systemof equations with no constant terms is called a homogeneous system of linear equations. A homogeneous linear system, in other words, has the following form:

$$\begin{gathered} a_{11}{x}_1+a_{12}{x}_2+\dots +a_{1n}{x}_n=0 \\ {a}_{21}{x}_1+a_{22}{x}_2+\dots +a_{2n}{x}_n=0 \\ {a}_{m1}{x}_1+a_{m2}{x}_2+\dots +a_{mn}{x}_n=0 \end{gathered}$$

For examples:

$\left\{\begin{array}{l}\text{2}x-\text{5}y=\text{0}\\ x-\text{2}y=\text{0}\end{array}\right.$is a homogeneous system in two variables.

role="math" localid="1664262035045" $\left\{\begin{array}{c}x+y+z=\text{0}\\ y-z=\text{0}\\ x+\text{2}y=\text{0}\end{array}\right.$is a homogeneous system in three variables.

The given Homogeneous equations are $\left\{\begin{array}{c}\text{2}x+\text{3}z=\text{0}\\ \text{4}x+\text{2}y+\text{5}z=\text{0}\\ x-y+\text{2}z=\text{0}\end{array}\right.$.

Find the sets of homogeneous equations with the help of the row reduction method.

Convert the given equations into the matrix form.

$\left[\begin{array}{cccc}2& 0& 3& 0\\ 4& 2& 5& 0\\ 1& -1& 2& 0\end{array}\right]$

Divide row 1 by 2: ${\text{R}}_1=\frac{{\text{R}}_1}{2}$.

$\left[\begin{array}{cccc}2& 0& \frac{3}{2}& 0\\ 4& 2& 5& 0\\ 1& -1& 2& 0\end{array}\right]$

Subtract row 1 multiplied by 4 from row 2: ${\text{R}}_{\text{2}}={\text{R}}_{\text{2}}-4{\text{R}}_{\text{1}}$.

$\left[\begin{array}{cccc}2& 0& \frac{3}{2}& 0\\ 0& 2& -1& 0\\ 1& -1& 2& 0\end{array}\right]$

Subtract row 1 from row 3: ${\text{R}}_{\text{3}}={\text{R}}_3-{\text{R}}_{\text{1}}$.

$\left[\begin{array}{cccc}2& 0& \frac{3}{2}& 0\\ 0& 2& -1& 0\\ 1& -1& \frac{1}{2}& 0\end{array}\right]$

Divide row 2 by 2: ${\text{R}}_2=\frac{{\text{R}}_2}{2}$.

$\left[\begin{array}{cccc}1& 0& \frac{3}{2}& 0\\ 0& 1& -\frac{1}{2}& 0\\ 1& -1& \frac{1}{2}& 0\end{array}\right]$

Add row 2 to row 3: ${\text{R}}_3={\text{R}}_3+{\text{R}}_2$.

role="math" localid="1664262316761" $\left[\begin{array}{cccc}1& 0& \frac{3}{2}& 0\\ 0& 1& -\frac{1}{2}& 0\\ 0& 0& 0& 0\end{array}\right]$

Therefore,$x=-\text{2}y\text{,}z=\text{2}y$ are the sets of given homogeneous equations.