To solve the given system of equations, we need to find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy both equations:
[tex]\[
\left\{\begin{array}{c}
-2a + 3b = 14 \\
a - 4b = 3
\end{array}\right.
\][/tex]
### Step-by-Step Solution
1. Solve for [tex]\(a\)[/tex] in the second equation:
[tex]\[
a - 4b = 3 \quad \Rightarrow \quad a = 4b + 3
\][/tex]
2. Substitute [tex]\(a = 4b + 3\)[/tex] into the first equation:
[tex]\[
-2(4b + 3) + 3b = 14
\][/tex]
3. Simplify the equation:
[tex]\[
-8b - 6 + 3b = 14
\][/tex]
[tex]\[
-5b - 6 = 14
\][/tex]
4. Solve for [tex]\(b\)[/tex]:
[tex]\[
-5b = 14 + 6
\][/tex]
[tex]\[
-5b = 20
\][/tex]
[tex]\[
b = -4
\][/tex]
5. Substitute [tex]\(b = -4\)[/tex] back into [tex]\(a = 4b + 3\)[/tex] to find [tex]\(a\)[/tex]:
[tex]\[
a = 4(-4) + 3
\][/tex]
[tex]\[
a = -16 + 3
\][/tex]
[tex]\[
a = -13
\][/tex]
### Verification
We have found that [tex]\((a, b) = (-13, -4)\)[/tex]. Let's verify this solution by substituting these values back into the original equations:
1. Substitute into the first equation:
[tex]\[
-2(-13) + 3(-4) = 14
\][/tex]
[tex]\[
26 - 12 = 14
\][/tex]
[tex]\[
14 = 14 \quad \text{(True)}
\][/tex]
2. Substitute into the second equation:
[tex]\[
-13 - 4(-4) = 3
\][/tex]
[tex]\[
-13 + 16 = 3
\][/tex]
[tex]\[
3 = 3 \quad \text{(True)}
\][/tex]
The values satisfy both equations.
### Conclusion
The ordered pair [tex]\((-13, -4)\)[/tex] is a solution to the system of equations. Among the given choices:
[tex]\[
(-13, -4)
\][/tex]
[tex]\[
(-4, -13)
\][/tex]
[tex]\[
\left(16 \frac{10}{11}, \frac{8}{11}\right)
\][/tex]
[tex]\[
(-4,13)
\][/tex]
The correct choice is the first one, [tex]\((-13, -4)\)[/tex]. Therefore, the answer is:
[tex]\[
\boxed{1}
\][/tex]
