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.\\a. there is one solution. the solution set is \\{\\(\boldsymbol{-1}, \boldsymbol{-2}, \boldsymbol{-7}\\)\\}. (simplify your answers.)\\b. there are infinitely many solutions. the system is dependent.\\c. there is no solution.

Explanation:

Step1: Simplify the first equation

Divide the first equation \(3x - 6y + 3z=-12\) by 3, we get \(x - 2y+z = - 4\).

Step2: Add the simplified first equation and the second equation

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

Step3: Subtract the second equation from the third equation

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

Step4: Solve the system of Equation 4 and Equation 5

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

Step5: Find y using x = - 1

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

Step6: Find z using x and y

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

Answer:

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