Q. 52 - 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 = -4

2x - 3y + 4z = -15

5x + y - 2z = 12

Step-by-Step Solution

Answer

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

Step 1: Given information

We are given a system of equation

x - y + z = -4                      (1)

2x - 3y + 4z = -15              (2)

5x + y - 2z = 12                 (3)

Step 2: Add equation 1 and 3

We get,

$x-y+z=-4+5x+y-2z=126x-z=8$

We get $6x-z=8 \left(4\right)$

Now multiply equation 3 by 3 and add it to equation 2 

We get,

$$\begin{gathered} 3(5x+y-2z=12) \\ 15x+3y-6z=36 \end{gathered}$$             

Now we add the equations

$2x-3y+4x=-15+15x+3y-6z=3617x-2z=21$

We get, $17x-2z=21$         (5)

Step 3: Multiply equation 4 by 2 and subtract equation 5 from it

We get,

$$\begin{gathered} 2(6x-z=8) \\ 12x-2z=16 \end{gathered}$$

Now we subtract from equation 5 from it

$12x-2z=16-17x+2z=-21-5x=-5$

Hence $x=1$

Step 4: Find the values of y and z

We get,

$$\begin{gathered} 6x-z=8 \\ 6-z=8 \\ z=-2 \end{gathered}$$

Similarly

$$\begin{gathered} x-y+z=-4 \\ 1-y-2=-4 \\ y=3 \end{gathered}$$

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