To find the ordered pair [tex]\((a_1, b)\)[/tex] that is the solution to the system of equations:
[tex]\[
\left\{\begin{array}{c}
-2a + 3b = 14 \\
a - 4b = 3
\end{array}\right.
\][/tex]
we will solve this system using the method of substitution or elimination. Here’s a step-by-step solution for the problem:
### Step 1: Write the System of Equations
We have the following system:
1. [tex]\(-2a + 3b = 14\)[/tex]
2. [tex]\(a - 4b = 3\)[/tex]
### Step 2: Solve One of the Equations for One Variable
We can solve the second equation for [tex]\(a\)[/tex]:
[tex]\[
a - 4b = 3 \implies a = 3 + 4b
\][/tex]
### Step 3: Substitute the Expression into the Other Equation
Substitute [tex]\(a = 3 + 4b\)[/tex] into the first equation:
[tex]\[
-2(3 + 4b) + 3b = 14
\][/tex]
### Step 4: Simplify and Solve for [tex]\(b\)[/tex]
Distribute and combine like terms:
[tex]\[
-6 - 8b + 3b = 14
\][/tex]
[tex]\[
-6 - 5b = 14
\][/tex]
Add 6 to both sides:
[tex]\[
-5b = 20
\][/tex]
Divide by -5:
[tex]\[
b = -4
\][/tex]
### Step 5: Substitute Back to Find [tex]\(a\)[/tex]
Substitute [tex]\(b = -4\)[/tex] back into the equation [tex]\(a = 3 + 4b\)[/tex]:
[tex]\[
a = 3 + 4(-4)
\][/tex]
[tex]\[
a = 3 - 16
\][/tex]
[tex]\[
a = -13
\][/tex]
### Step 6: Verify the Solution
To ensure our solution [tex]\((a, b) = (-13, -4)\)[/tex] satisfies both equations, we can substitute these values back into the original equations:
First equation:
[tex]\[
-2(-13) + 3(-4) = 26 - 12 = 14
\][/tex]
Second equation:
[tex]\[
-13 - 4(-4) = -13 + 16 = 3
\][/tex]
Both equations are satisfied.
### Step 7: Identify the Correct Ordered Pair
Given the options:
- [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 ordered pair solution that matches our calculated values is [tex]\((-13, -4)\)[/tex].
Therefore, the ordered pair [tex]\((a_1, b)\)[/tex] that solves the system of equations is:
[tex]\[
\boxed{(-13, -4)}
\][/tex]
