Question

solve the system.\

$$\begin{cases}5x + 4y - 5z = 7\\\\2x - 5y + 2z = -1\\\\4x - 2y + 4z = 22\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\\(\bigcirc\\) a. there is one solution. the solution set is \\(\\{ (\square, \square, \square) \\}\\). (simplify your answers.)\\(\bigcirc\\) b. there are infinitely many solutions. the system is dependent.\\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Simplify the third equation

Divide the third equation \(4x - 2y + 4z = 22\) by 2, we get \(2x - y + 2z = 11\).

Step2: Subtract the second equation from the new third equation

The second equation is \(2x - 5y + 2z = -1\), subtract it from \(2x - y + 2z = 11\):
\[

$$\begin{align*} (2x - y + 2z)-(2x - 5y + 2z)&=11-(-1)\\ 2x - y + 2z - 2x + 5y - 2z&=12\\ 4y&=12\\ y&=3 \end{align*}$$

\]

Step3: Substitute \(y = 3\) into the second and new third equations

Substitute \(y = 3\) into \(2x - 5y + 2z = -1\), we have \(2x-15 + 2z=-1\), which simplifies to \(2x + 2z=14\), or \(x + z = 7\) (Equation 1).
Substitute \(y = 3\) into \(2x - y + 2z = 11\), we get \(2x-3 + 2z=11\), which simplifies to \(2x + 2z=14\), same as above.
Substitute \(y = 3\) into the first equation \(5x + 4y - 5z = 7\), we get \(5x+12 - 5z=7\), which simplifies to \(5x - 5z=-5\), or \(x - z=-1\) (Equation 2).

Step4: Solve the system of Equation 1 and Equation 2

We have the system \(

$$\begin{cases}x + z = 7\\x - z=-1\end{cases}$$

\)
Add the two equations:
\[

$$\begin{align*} (x + z)+(x - z)&=7+(-1)\\ 2x&=6\\ x&=3 \end{align*}$$

\]
Substitute \(x = 3\) into \(x + z = 7\), we get \(3+z = 7\), so \(z = 4\).

Answer:

A. There is one solution. The solution set is \(\{(3, 3, 4)\}\)