Question

In Problems 75– 8 2, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}x-2y+4z=2\\ -3x+5y-2z=17\\ 4x-3y=-22\end{array}\right.$

Short answer

The values of unknown variables are $x=-4,y=2$and$z=2.5$

Step-by-step solution

Step 1. Given information

The given equations$\left\{\begin{array}{l}x-2y+4z=2\\ -3x+5y-2z=17\\ 4x-3y=-22\end{array}\right.$

Step 2. Finding the values of x,y

Consider the equations $x-2y+4z=2,-3x+5y-2z=17$and $4x-3y=-22$.To solve these equations,we first convert them into matrix form and then transform it into reduced echelon form

$$\begin{gathered} \left[\begin{array}{ccc}1& -2& 4\\ -3& 5& -2\\ 4& -3& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 17\\ -22\end{array}\right]\left\{R_2=R_2+3R_1;R_3=R_3-4R_1\right\} \\ \left[\begin{array}{ccc}1& -2& 4\\ 0& -1& 10\\ 0& 5& -16\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 23\\ -30\end{array}\right]\left\{R_2=-R_2\right\} \\ \left[\begin{array}{ccc}1& 0& 4\\ 0& 1& -10\\ 0& 0& -16\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ -23\\ -30\end{array}\right]\left\{R_1=R_1+2R_2;R_3=R_3-5R_2\right\} \\ \left[\begin{array}{ccc}1& 0& -16\\ 0& 1& -10\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-44\\ -23\\ 85\end{array}\right]\left\{R_3=\frac{R_3}{34}\right\} \\ \left[\begin{array}{ccc}1& 0& -16\\ 0& 1& -10\\ 0& 0& -1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-44\\ -23\\ 2.5\end{array}\right]\left\{R_1=R_1+16R_3;R_2=R_2+10R_3\right\} \\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 2.5\end{array}\right] \\ \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 2.5\end{array}\right] \end{gathered}$$

Thus,we have solved the equations and the values of unknown variables are$x=-4,y=2$ and$z=2.5$