Q. 3 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 1–4, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l} x-y+2z=5\\ 3x+4y-z=-2\\ 5x+2y+3z=8\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of equations $\left\{\begin{array}{l} x-y+2z=5\\ 3x+4y-z=-2\\ 5x+2y+3z=8\end{array}\right.$ is $(\frac{1-7y}{7},y,\frac{17+7y}{7})$

Step 1. Given

The system of equations:

$\left\{\begin{array}{l} x-y+2z=5\\ 3x+4y-z=-2\\ 5x+2y+3z=8\end{array}\right.$

Step 2. Solve the system

Subtract the second equation from first equation,

$-2x-5y+3z=7$

Subtract the above equation from third equation,

$7x+7y=1$

Step 3. Multiply third equation by 2

Multiply second equation by 2,

$6x+8y-2z=-4$

Subtract first equation from above equation,

$7x+7y=1$

Step 4. Solve the system

When the equation is reduced to two variables, the resulting equations are same.

So the system of equation with three variables must be reduced to two.

Consider $7x+7y=1$

                         $x=\frac{1-7y}{7}$

Step 5. Find y

Substitute $x=\frac{1-7y}{7}$ in the first equation to get the value of $z$ in terms of $y$,

$$\begin{gathered} x-y+2z=5 \\ \frac{1-7y}{7}-y+2z=5 \\ \frac{1-14y}{7}+2z=5 \\ 2z=5-\frac{1-14y}{7} \\ 2z=\frac{34+14y}{7} \\ z=\frac{17+7y}{7} \end{gathered}$$