Let's solve the system of linear equations step-by-step to find the ordered pair [tex]\((a, b)\)[/tex].
The system of equations given is:
[tex]\[
\left\{
\begin{aligned}
-2a + 3b &= 14 \\
a - 4b &= 3
\end{aligned}
\right.
\][/tex]
Step 1: Solve the second equation for [tex]\(a\)[/tex] in terms of [tex]\(b\)[/tex]
Starting with the equation:
[tex]\[
a - 4b = 3
\][/tex]
We can express [tex]\(a\)[/tex] as:
[tex]\[
a = 3 + 4b
\][/tex]
Step 2: Substitute [tex]\(a\)[/tex] into the first equation
Now we substitute [tex]\(a = 3 + 4b\)[/tex] into the first equation [tex]\(-2a + 3b = 14\)[/tex]:
[tex]\[
-2(3 + 4b) + 3b = 14
\][/tex]
Step 3: Simplify the equation
Distribute and combine like terms:
[tex]\[
-6 - 8b + 3b = 14
\][/tex]
[tex]\[
-6 - 5b = 14
\][/tex]
Step 4: Solve for [tex]\(b\)[/tex]
Add 6 to both sides:
[tex]\[
-5b = 20
\][/tex]
Divide by -5:
[tex]\[
b = -4
\][/tex]
Step 5: Substitute [tex]\(b\)[/tex] back into the expression for [tex]\(a\)[/tex]
Using [tex]\(a = 3 + 4b\)[/tex] and substituting [tex]\(b = -4\)[/tex]:
[tex]\[
a = 3 + 4(-4)
\][/tex]
[tex]\[
a = 3 - 16
\][/tex]
[tex]\[
a = -13
\][/tex]
Thus, the solution to the system of equations is the ordered pair:
[tex]\[
(a, b) = (-13, -4)
\][/tex]
Step 6: Verify the solution with the given options
Given options are:
1. [tex]\((-13, -4)\)[/tex]
2. [tex]\((-13, 4)\)[/tex]
3. [tex]\((-4, -13)\)[/tex]
4. [tex]\((-4, 13)\)[/tex]
From our solution, the correct ordered pair is [tex]\((-13, -4)\)[/tex].
Therefore, the index of the correct option is:
[tex]\[
0
\][/tex]
