Q. 53 - Precalculus Enhanced with Graphing Utilities

Question

Solve each system of equations. If the system has no solution, say that it is inconsistent  

x + 2y - z = -3

2x - 4y + z = -7

-2x + 2y - 3z = 4

Step-by-Step Solution

Answer

The solution set of given system is $\left\{\right(x,y,z)=(-3,\frac{1}{2},1\left)\right\}$

Step 1: Given information

We are given a system of equation

$$\begin{gathered} x+ 2y-z=-3 \left(1\right) \\ 2x- 4y+z=-7 \left(2\right) \\ -2x+2y-3z=4 \left(3\right) \end{gathered}$$

Step 2: Add equation 2 and 3

We get,

$2x-4y+z=-7+-2x+2y-3z=4-2y-2z=-3$

Hence we get $-2y-2x=3 \left(4\right)$

Multiply equation 1 by 2 and add it to the equation 3

We get,

$$\begin{gathered} 2(x+2y-z=-3) \\ 2x+4y-2z=-6 \end{gathered}$$

Also

$2x+4y-2z=-6+-2x+2y-3z=46y-5z=-2$

Hence $6y-5z=-2 \left(5\right)$

Step 3: Multiply equation 4 by 3 and add it to equation 5

We get,

$$\begin{gathered} 3(-2y-2z=-3) \\ -6y-6z=-9 \end{gathered}$$

Also

$6y-5z=-2+-6y-6z=-9-11z=-11$

Hence $z=1$

Step 4: Find the values of x and y

We get,

$$\begin{gathered} 2y+2z=3 \\ 2y+2=3 \\ 2y=1 \\ y=\frac{1}{2} \end{gathered}$$

Also

$$\begin{gathered} x+2y-z=-3 \\ x+1-1=-3 \\ x=-3 \end{gathered}$$

The solution set is $\left\{\right(x,y,z)=(-3,\frac{1}{2},1\left)\right\}$