Q. 51 - Precalculus Enhanced with Graphing Utilities

Question

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

x + y - z = 6

3x - 2y + z = -5

x + 3y - 2z = 14

Step-by-Step Solution

Answer

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

Step 1: Given information

We are given a system of equations

x + y - z = 6                      (1)

3x - 2y + z = -5                (2)

x + 3y - 2z = 14                (3)

Step 2: Subtract equation 1 and 3

We get,

$x+y-z=6-x-3y+2z=-14-2y+z=-8$

We get $-2y+z=-8$             (4)

Step 3: Multiply equation 1 by 3 and then subtract from equation 2

We get,

$$\begin{gathered} 3(x+y-z=6) \\ 3x+3y-3z=18 \left(5\right) \end{gathered}$$

Now subtract from equation (4)

$3x+3y-3z=18-3x+2y-z=55y-4z=23$

hence we get, $5y-4z=23$     (6)

Step 4: Multiply equation 4 by 4 And add equation 6

We get,

$$\begin{gathered} 4(-2y+z=-8) \\ -8y+4z=-32 \end{gathered}$$

Now we add equation 6 to it

$-8y+4z=-32+5y-4z=23-3y=-9$

hence $y=3$

Step 5: Find the values of x and z

We have,

$$\begin{gathered} -2y+z=-8 \\ -2\left(3\right)+z=-8 \\ -6+z=-8 \\ z=-2 \end{gathered}$$

Now substitute the values of y and z in equation 1

$$\begin{gathered} x+y-z=6 \\ x+3-(-2)=6 \\ x+3+2=6 \\ x=1 \end{gathered}$$

hence solution of the system is (1,3,-2).