Use the substitution method to solve the system. If the equations of the system are dependent, or if the system is inconsistent, so indicate.
E) The system is inconsistent.

Question

Use the substitution method to solve the system. If the equations of the system are dependent, or if the system is inconsistent, so indicate. 
E) The system is inconsistent.

Quizplus · Accepted Answer

The system is:

\begin{align*}
x+2y-z &= -1 \
3x+y+4z &= 9 \
-2x+3y+5z &= 2
\end{align*}

Solving the first equation for $x$ gives $x = -2y+z-1$. Substituting this expression for $x$ in the second equation gives:

$$3(-2y+z-1)+y+4z = 9$$

Simplifying gives:

$$-5y+7z = 12$$

Solving the second equation for $y$ gives $y = -\frac{7}{5}z+\frac{12}{5}$. Substituting this expression for $y$ and the expression for $x$ in the third equation gives:

$$-2(-2y+z-1)+3\left(-\frac{7}{5}z+\frac{12}{5}ight)+5z = 2$$

Simplifying gives:

$$z = 2$$

Back-substituting gives:

\begin{align*}
x &= -2y+z-1 = -2\left(-\frac{7}{5}(2)+\frac{12}{5}ight)+2-1 = 3 \
y &= -\frac{7}{5}z+\frac{12}{5} = -\frac{7}{5}(2)+\frac{12}{5} = -\frac{2}{5}
\end{align*}

Therefore, the solution to the system is $(x,y,z) = (3,-\frac{2}{5},2)$, and the system is consistent and independent. Choice D is the correct answer.

Use the Substitution Method to Solve the System