To solve the system of equations below:
[tex]\[
\begin{cases}
3x + y = -18 \\
5x + 3y = 2
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously. Here is the step-by-step process:
1. Equation 1:
[tex]\[ 3x + y = -18 \][/tex]
2. Equation 2:
[tex]\[ 5x + 3y = 2 \][/tex]
Let’s start by isolating [tex]\( y \)[/tex] in the first equation:
[tex]\[ y = -18 - 3x \][/tex]
Next, we substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[ 5x + 3(-18 - 3x) = 2 \][/tex]
Simplify the equation:
[tex]\[ 5x - 54 - 9x = 2 \][/tex]
Combine like terms:
[tex]\[ -4x - 54 = 2 \][/tex]
Add 54 to both sides:
[tex]\[ -4x = 56 \][/tex]
Divide both sides by -4:
[tex]\[ x = -14 \][/tex]
Now that we have [tex]\( x = -14 \)[/tex], we substitute this back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = -18 - 3(-14) \][/tex]
Calculate the result:
[tex]\[ y = -18 + 42 \][/tex]
[tex]\[ y = 24 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = -14 \][/tex]
[tex]\[ y = 24 \][/tex]
So, the solution is [tex]\((-14, 24)\)[/tex], which corresponds to one of the given choices.
