To find the ordered pair [tex]\((c, d)\)[/tex] that solves the given system of linear equations:
[tex]\[
\left\{\begin{aligned}
-c + 2d &= 13 \\
-9c - 4d &= -15
\end{aligned}\right.
\][/tex]
we need to determine the correct pair from the options provided: [tex]\((-1, 6)\)[/tex], [tex]\((-1, 7)\)[/tex], [tex]\((6, -1)\)[/tex], and [tex]\((7, 10)\)[/tex].
1. Write the first equation as:
[tex]\[
-c + 2d = 13 \quad \text{(1)}
\][/tex]
2. Write the second equation as:
[tex]\[
-9c - 4d = -15 \quad \text{(2)}
\][/tex]
3. Let's test each ordered pair to see which one satisfies both equations:
- Option [tex]\((-1, 6)\)[/tex]:
[tex]\[
\begin{aligned}
-(-1) + 2(6) &= 1 + 12 = 13 \quad \text{(satisfies equation 1)} \\
-9(-1) - 4(6) &= 9 - 24 = -15 \quad \text{(satisfies equation 2)}
\end{aligned}
\][/tex]
Therefore, [tex]\((-1, 6)\)[/tex] satisfies both equations.
- Option [tex]\((-1, 7)\)[/tex]:
[tex]\[
\begin{aligned}
-(-1) + 2(7) &= 1 + 14 = 15 \quad \text{(does not satisfy equation 1)} \\
\end{aligned}
\][/tex]
Since [tex]\((-1, 7)\)[/tex] does not satisfy the first equation, it is not the correct solution.
- Option [tex]\((6, -1)\)[/tex]:
[tex]\[
\begin{aligned}
-(6) + 2(-1) &= -6 - 2 = -8 \quad \text{(does not satisfy equation 1)} \\
\end{aligned}
\][/tex]
Since [tex]\((6, -1)\)[/tex] does not satisfy the first equation, it is not the correct solution.
- Option [tex]\((7, 10)\)[/tex]:
[tex]\[
\begin{aligned}
-(7) + 2(10) &= -7 + 20 = 13 \quad \text{(satisfies equation 1)} \\
-9(7) - 4(10) &= -63 - 40 = -103 \quad \text{(does not satisfy equation 2)}
\end{aligned}
\][/tex]
Since [tex]\((7, 10)\)[/tex] does not satisfy the second equation, it is not the correct solution.
After evaluating each option, we find that the ordered pair [tex]\((-1, 6)\)[/tex] is the only one that satisfies both equations in the given system of linear equations.
Therefore, the correct ordered pair is:
[tex]\[
\boxed{(-1, 6)}
\][/tex]
