To solve the system of linear equations:
[tex]\[
\left\{
\begin{array}{c l}
-c + 2d = 13 & \quad (1) \\
-9c - 4d = -15 & \quad (2)
\end{array}
\right.
\][/tex]
we can use the substitution or elimination method. Here, we will use the substitution method. First, solve equation (1) for [tex]\(c\)[/tex]:
### Step 1: Solve equation (1) for [tex]\(c\)[/tex]
[tex]\[
-c + 2d = 13
\][/tex]
[tex]\[
-c = 13 - 2d
\][/tex]
[tex]\[
c = 2d - 13
\][/tex]
### Step 2: Substitute [tex]\(c\)[/tex] into equation (2)
Substitute [tex]\(c = 2d - 13\)[/tex] into equation (2):
[tex]\[
-9(2d - 13) - 4d = -15
\][/tex]
[tex]\[
-18d + 117 - 4d = -15
\][/tex]
Combine like terms:
[tex]\[
-22d + 117 = -15
\][/tex]
### Step 3: Solve for [tex]\(d\)[/tex]
[tex]\[
-22d = -15 - 117
\][/tex]
[tex]\[
-22d = -132
\][/tex]
[tex]\[
d = \frac{-132}{-22}
\][/tex]
[tex]\[
d = 6
\][/tex]
### Step 4: Substitute back to find [tex]\(c\)[/tex]
Substitute [tex]\(d = 6\)[/tex] back into equation for [tex]\(c\)[/tex]:
[tex]\[
c = 2d - 13
\][/tex]
[tex]\[
c = 2(6) - 13
\][/tex]
[tex]\[
c = 12 - 13
\][/tex]
[tex]\[
c = -1
\][/tex]
### Step 5: Verify the solution in both equations
Substitute [tex]\(c = -1\)[/tex] and [tex]\(d = 6\)[/tex] back into the original equations to verify:
Equation (1):
[tex]\[
-c + 2d = 13
\][/tex]
[tex]\[
-(-1) + 2(6) = 13
\][/tex]
[tex]\[
1 + 12 = 13
\][/tex]
[tex]\[
13 = 13 \quad \text{(True)}
\][/tex]
Equation (2):
[tex]\[
-9c - 4d = -15
\][/tex]
[tex]\[
-9(-1) - 4(6) = -15
\][/tex]
[tex]\[
9 - 24 = -15
\][/tex]
[tex]\[
-15 = -15 \quad \text{(True)}
\][/tex]
Both equations hold true. Therefore, the ordered pair [tex]\((c, d) = (-1, 6)\)[/tex] is the solution to the system.
Thus, the correct answer is:
[tex]\[
(-1, 6)
\][/tex]
