Question

solve the system of equations by the addition method.\

$$\begin{cases}6x - y = -6\\\\9x + 3y = -117\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\boldsymbol{\circ}\\) a. the solution is \\(\square\\). (simplify your answer. type an ordered pair.)\\(\boldsymbol{\circ}\\) b. there are infinitely many solutions.\\(\boldsymbol{\circ}\\) c. there is no solution.

Explanation:

Step1: Multiply first equation by 3

To eliminate \( y \), we multiply the first equation \( 6x - y = -6 \) by 3. This gives us \( 3(6x - y)=3(-6) \), which simplifies to \( 18x - 3y = -18 \).

Step2: Add the two equations

Now we add the new equation \( 18x - 3y = -18 \) to the second equation \( 9x + 3y = -117 \).
\[

$$\begin{align*} (18x - 3y)+(9x + 3y)&=-18+(-117)\\ 18x - 3y+9x + 3y&=-135\\ 27x&=-135 \end{align*}$$

\]

Step3: Solve for \( x \)

Divide both sides of \( 27x = -135 \) by 27: \( x=\frac{-135}{27}=-5 \).

Step4: Substitute \( x \) into first equation

Substitute \( x = -5 \) into the first original equation \( 6x - y = -6 \): \( 6(-5)-y=-6 \), which simplifies to \( -30 - y = -6 \).

Step5: Solve for \( y \)

Add 30 to both sides: \( -y=-6 + 30=24 \), then multiply both sides by -1: \( y=-24 \).

Answer:

A. The solution is \((-5, -24)\)