Which ordered pair [tex]\((a, b)\)[/tex] is a solution to the given system of linear equations?

[tex]\[
\left\{
\begin{aligned}
-2a + 3b &= 14 \\
a - 4b &= 3
\end{aligned}
\right.
\][/tex]

\begin{tabular}{|l|}
\hline
[tex]\(( -13, -4 )\)[/tex] \\
[tex]\(( -13, 4 )\)[/tex] \\
[tex]\(( -4, -13 )\)[/tex] \\
[tex]\(( -4, 13 )\)[/tex] \\
\hline
\end{tabular}



Answer :

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]