To determine the correct ordered pair [tex]\((a, b)\)[/tex] that solves the given system of equations, we need to solve the following system:
[tex]\[
\left\{
\begin{array}{c}
-2 a + 3 b = 14 \\
a - 4 b = 3
\end{array}
\right.
\][/tex]
Firstly, we can solve this system using the method of substitution or elimination. Here’s a detailed step-by-step solution using the elimination method:
1. First, we take the two equations:
[tex]\[
-2a + 3b = 14 \quad \text{(1)}
\][/tex]
[tex]\[
a - 4b = 3 \quad \text{(2)}
\][/tex]
2. Multiply equation (2) by 2 so that the coefficient of [tex]\(a\)[/tex] matches:
[tex]\[
2 \cdot (a - 4b) = 2 \cdot 3
\][/tex]
[tex]\[
2a - 8b = 6 \quad \text{(3)}
\][/tex]
3. Add equation (1) to equation (3) to eliminate [tex]\(a\)[/tex]:
[tex]\[
(-2a + 3b) + (2a - 8b) = 14 + 6
\][/tex]
[tex]\[
(-2a + 2a) + (3b - 8b) = 20
\][/tex]
[tex]\[
-5b = 20
\][/tex]
4. Solve for [tex]\(b\)[/tex]:
[tex]\[
b = \frac{20}{-5} = -4
\][/tex]
5. Substitute [tex]\(b = -4\)[/tex] back into equation (2) to find [tex]\(a\)[/tex]:
[tex]\[
a - 4(-4) = 3
\][/tex]
[tex]\[
a + 16 = 3
\][/tex]
[tex]\[
a = 3 - 16
\][/tex]
[tex]\[
a = -13
\][/tex]
So, the solution to the system of equations is [tex]\((-13, -4)\)[/tex].
Comparing this solution to the given options, we see that the correct ordered pair [tex]\((a, b)\)[/tex] that solves the system of equations is:
[tex]\[
\boxed{(-13, -4)}
\][/tex]
