Q. 42 - Precalculus Enhanced with Graphing Utilities

Question

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

$$\begin{gathered} 2x+y=4 \\ -2y+4z=0 \\ 3x-2z=-11 \end{gathered}$$

Step-by-Step Solution

Answer

The solution of the system of equation is

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

Step 1: Given information

We are given a system of equation

$$\begin{gathered} 2x+y=4 \\ -2y+4z=0 \\ 3x-2z=-11 \end{gathered}$$

Step 2: multiply equation 1 by 2 and add it to the equation 2

We get,

$4x+2y=-8$

And now

$4x+2y=-8+-2y+4z=04x+4z=-8$

Step 3: Now we Multiply equation 3 by 2 and add it the new equation

We get,

$4x+4z=-8+6x-4z=-2210x=-30$

And hence $x=-3$

Step 4: Substitute the value of x to find the value of y,z

We get,

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

Similarly

$$\begin{gathered} 3x-2z=-11 \\ 3(-3)-2z=-11 \\ -9-2z=-11 \\ -2z=-2 \\ z=1 \end{gathered}$$

Step 5: Conclusion

The solution of the system of equation is $x=-3,y=2,z=1$