Q. 43 - Precalculus Enhanced with Graphing Utilities

Question

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

$$\begin{gathered} x-2y+3z=7 \\ 2x+y+z=4 \\ -3x+2y-2z=-10 \end{gathered}$$

Step-by-Step Solution

Answer

The solution of the system of equation is $x=2,y=-1,z=1$

Step 1: Given information

We are given a system of equation

$$\begin{gathered} x-2y+3z=7 \\ 2x+y+z=4 \\ -3x+2y-2z=-10 \end{gathered}$$

Step 2: Multiply equation 2 by 2

We get,

$4x+2y+2z=8$              (4)

Now add this to equation 2

We get,

$x-2y+3z=7+4x+2y+2z=85x+5z=15$

Hence 

$$\begin{gathered} 5x+5z=15 \\ x+z=3 \end{gathered}$$              (5)

Now subtract equation (4) from equation 3

$-3x+2y-2z=-10-4x-2y-2z=-8-7x-4z=-18$ 

$-7x-4z=-18$     (6)

Step 3: Find the value of x

Multiply equation 5 by 4 And add it to equation 6

We get,

$4x+4z=12$

Now we add this to equation 6

We get,

$4x+4z=12-7x-4z=-18-3x=-6$

hence $x=2$

Step 4: Find the value of y and z

We get,

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

And

$$\begin{gathered} x-2y+3z=7 \\ 2-2y+3=7 \\ -2y=2 \\ y=-1 \end{gathered}$$

Step 5: Conclusion

The solution of the system of equation is $x=2,y=-1,z=1$