Question

solve the system.\

$$\begin{cases}3x - 6y + 3z = -12\\\\x + 3y - z = 0\\\\6x - y - z = 3\\end{cases}$$

\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 first equation

Divide the first equation \(3x - 6y + 3z = -12\) by 3, we get \(x - 2y + z = -4\). Let's call this Equation (4).

Step2: Add Equation (4) and the second equation

The second equation is \(x + 3y - z = 0\) (Equation 2). Add Equation (4) and Equation (2): \((x - 2y + z)+(x + 3y - z)=-4 + 0\), which simplifies to \(2x + y = -4\) (Equation 5).

Step3: Subtract the second equation from the third equation

The third equation is \(6x - y - z = 3\) (Equation 3). Subtract Equation (2) from Equation (3): \((6x - y - z)-(x + 3y - z)=3 - 0\), which simplifies to \(5x - 4y = 3\) (Equation 6).

Step4: Solve the system of Equation (5) and Equation (6)

From Equation (5): \(y = -4 - 2x\). Substitute into Equation (6): \(5x - 4(-4 - 2x)=3\). Expand: \(5x + 16 + 8x = 3\), combine like terms: \(13x = 3 - 16=-13\), so \(x = -1\).

Step5: Find y using x = -1

Substitute \(x = -1\) into Equation (5): \(2(-1)+y = -4\), so \(-2 + y = -4\), then \(y = -2\).

Step6: Find z using x and y

Substitute \(x = -1\) and \(y = -2\) into Equation (2): \(-1 + 3(-2)-z = 0\), which is \(-1 - 6 - z = 0\), so \(-7 - z = 0\), then \(z = -7\).

Answer:

A. There is one solution. The solution set is \(\{(-1, -2, -7)\}\).