$$ 1 \leftarrow \begin{array}{l} ext { Solve the given system of equations. } \ \begin{array}{l}5 x+4 y-5 z=-21 \ 3 x-4 y+3 z=-7 \ 6 x-3 y+6 z=-9\end{array} \ ext { Select the correct choice below and fill in any answer boxes within your choice. } \ ext { A. There is one solution. The solution set is }\{(\square, \square)\} ext {. (Simplify your answers.) } \ ext { B. There are infinitely many solutions. } \ ext { C. There is no solution. }\end{array} $$

Question

$$ 1 \leftarrow \begin{array}{l}	ext { Solve the given system of equations. } \ \begin{array}{l}5 x+4 y-5 z=-21 \ 3 x-4 y+3 z=-7 \ 6 x-3 y+6 z=-9\end{array} \ 	ext { Select the correct choice below and fill in any answer boxes within your choice. } \ 	ext { A. There is one solution. The solution set is }\{(\square, \square)\} 	ext {. (Simplify your answers.) } \ 	ext { B. There are infinitely many solutions. } \ 	ext { C. There is no solution. }\end{array} $$

UpStudy AI · Accepted Answer

$$
\textbf{Step 1. Write the system of equations:}
$$
$$
\begin{aligned}
\text{(1)} \quad 5x + 4y - 5z &= -21, \
\text{(2)} \quad 3x - 4y + 3z &= -7, \
\text{(3)} \quad 6x - 3y + 6z &= -9.
\end{aligned}
$$

$$
\textbf{Step 2. Eliminate } y \textbf{ by adding equations (1) and (2):}
$$
$$
(5x + 4y - 5z) + (3x - 4y + 3z) = -21 - 7.
$$
$$
8x - 2z = -28.
$$
$$
\text{Divide by } 2: \quad 4x - z = -14.
$$
$$
\text{Solve for } z: \quad z = 4x + 14.
$$

$$
\textbf{Step 3. Substitute } z = 4x + 14 \textbf{ into equation (2):}
$$
$$
3x - 4y + 3(4x+14) = -7.
$$
$$
3x - 4y + 12x + 42 = -7.
$$
$$
15x - 4y = -49.
$$
$$
\text{Solve for } y:
$$
$$
-4y = -49 - 15x \quad \Longrightarrow \quad y = \frac{49 + 15x}{4}.
$$

$$
\textbf{Step 4. Substitute } z = 4x + 14 \textbf{ into equation (3):}
$$
$$
6x - 3y + 6(4x+14) = -9.
$$
$$
6x - 3y + 24x + 84 = -9.
$$
$$
30x - 3y = -93.
$$
$$
\text{Divide by } 3: \quad 10x - y = -31.
$$
$$
\text{Solve for } y: \quad y = 10x + 31.
$$

$$
\textbf{Step 5. Equate the two expressions for } y:
$$
$$
\frac{49 + 15x}{4} = 10x + 31.
$$
$$
\text{Multiply both sides by } 4:
$$
$$
49 + 15x = 40x + 124.
$$
$$
49 - 124 = 40x - 15x.
$$
$$
-75 = 25x.
$$
$$
x = \frac{-75}{25} = -3.
$$

$$
\textbf{Step 6. Find } y \textbf{ using } y = 10x + 31:
$$
$$
y = 10(-3) + 31 = -30 + 31 = 1.
$$

$$
\textbf{Step 7. Find } z \textbf{ using } z = 4x + 14:
$$
$$
z = 4(-3) + 14 = -12 + 14 = 2.
$$

$$
\textbf{Step 8. Write the solution set:}
$$
$$
\{(-3, 1, 2)\}
$$

$$
\textbf{Answer: Choice A. There is one solution. The solution set is } \{(-3, 1, 2)\}.
$$
There is one solution: $$ x = -3 $$, $$ y = 1 $$, $$ z = 2 $$.

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