To solve the system of equations:
[tex]\[
\begin{cases}
2x = 5y + 4 \\
3x - 2y = -16
\end{cases}
\][/tex]
we can use the method of substitution or elimination. Let's go step by step using substitution in this case.
1. Equation 1: [tex]\(2x = 5y + 4\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{5y + 4}{2}
\][/tex]
2. Substitute [tex]\(x\)[/tex] in Equation 2:
[tex]\[
3 \left( \frac{5y + 4}{2} \right) - 2y = -16
\][/tex]
3. Multiply both sides by 2 to clear the fraction:
[tex]\[
3(5y + 4) - 4y = -32
\][/tex]
4. Distribute and simplify:
[tex]\[
15y + 12 - 4y = -32
\][/tex]
[tex]\[
11y + 12 = -32
\][/tex]
5. Isolate [tex]\(y\)[/tex]:
[tex]\[
11y = -32 - 12
\][/tex]
[tex]\[
11y = -44
\][/tex]
[tex]\[
y = \frac{-44}{11}
\][/tex]
[tex]\[
y = -4
\][/tex]
6. Substitute [tex]\(y\)[/tex] back into Equation 1 to find [tex]\(x\)[/tex]:
[tex]\[
2x = 5(-4) + 4
\][/tex]
[tex]\[
2x = -20 + 4
\][/tex]
[tex]\[
2x = -16
\][/tex]
[tex]\[
x = \frac{-16}{2}
\][/tex]
[tex]\[
x = -8
\][/tex]
Therefore, the solution to the system of equations is [tex]\((-8, -4)\)[/tex].
To determine which option matches our solution:
- [tex]\((-8, -4)\)[/tex]: Yes, this matches our solution.
Thus, the correct answer is [tex]\((-8, -4)\)[/tex], which corresponds to the option 1.
