To solve the system of equations:
[tex]\[
\begin{array}{l}
x + 4y = 13 \\
2x + 3y = 6
\end{array}
\][/tex]
we will use one of the methods typically employed for solving systems of linear equations, such as substitution, elimination, or matrix methods. Here, I'll describe the process clearly.
### Method: Elimination
Step 1: Multiply the equations to make the coefficients of x (or y) the same in both equations, thereby allowing us to eliminate one of the variables.
Let's start by aligning the coefficients of [tex]\(x\)[/tex].
- Multiply the first equation by 2:
[tex]\[
2(x + 4y) = 2(13)
\][/tex]
[tex]\[
2x + 8y = 26
\][/tex]
Now we have:
[tex]\[
2x + 8y = 26 \quad \text{(Equation 3)}
\][/tex]
[tex]\[
2x + 3y = 6 \quad \text{(Equation 2)}
\][/tex]
Step 2: Subtract Equation 2 from Equation 3 to eliminate [tex]\(x\)[/tex]:
[tex]\[
(2x + 8y) - (2x + 3y) = 26 - 6
\][/tex]
[tex]\[
2x + 8y - 2x - 3y = 20
\][/tex]
[tex]\[
5y = 20
\][/tex]
[tex]\[
y = \frac{20}{5}
\][/tex]
[tex]\[
y = 4
\][/tex]
Step 3: Substitute [tex]\(y = 4\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. We'll use the first equation:
[tex]\[
x + 4(4) = 13
\][/tex]
[tex]\[
x + 16 = 13
\][/tex]
[tex]\[
x = 13 - 16
\][/tex]
[tex]\[
x = -3
\][/tex]
So, the solution to the system of equations is [tex]\(x = -3\)[/tex] and [tex]\(y = 4\)[/tex]. This gives us the point [tex]\((-3, 4)\)[/tex].
Step 4: Check the solution against the multiple choice answers:
1. [tex]\((3, 5)\)[/tex]
2. [tex]\((-4, 5)\)[/tex]
3. [tex]\((-3, 4)\)[/tex]
4. No solution
5. Infinitely many solutions
Our solution [tex]\((-3, 4)\)[/tex] matches option 3.
Therefore, the final answer is:
[tex]\[
\boxed{3}
\][/tex]
