To solve the given system of linear equations and find the correct solution, follow these detailed steps:
### Step 1: Write Down the System of Equations
We are given the following system of equations:
[tex]\[
\left\{
\begin{array}{l}
3.16x + 12.82y = -23.77 \\
7.95x - 4.58y = 87.71
\end{array}
\right.
\][/tex]
### Step 2: Solve the System of Equations
We can solve this system using various methods such as substitution, elimination, or matrix methods. Here, we'll outline the steps you might use:
#### Method 1: Substitution or Elimination (Concept Outline)
1. Isolate one variable from one of the equations if you're using substitution. For instance, solve for [tex]\(x\)[/tex] or [tex]\(y\)[/tex].
2. Substitute the expression from step 1 into the other equation and solve for the remaining variable.
3. Back-substitute the found value into one of the original equations to find the other variable.
#### Method 2: Matrix Method (Concept Outline)
1. Write the system in matrix form: [tex]\(Ax = b\)[/tex].
2. Invert the matrix [tex]\(A\)[/tex] if it is invertible and multiply by the vector [tex]\(b\)[/tex] to solve for [tex]\(x\)[/tex].
### Step 3: Verification of Possible Solutions
Given possible solutions to check:
1. [tex]\((8.69, -4.02)\)[/tex]
2. [tex]\((5.87, -3.45)\)[/tex]
3. [tex]\((-4.05, 1.21)\)[/tex]
4. [tex]\((-7.76, -1.66)\)[/tex]
Based on our calculations (conceptually detailed above), the precise solution for the system of equations is:
[tex]\[
x \approx 8.72549097101342
\][/tex]
[tex]\[
y \approx -4.00487920970378
\][/tex]
### Step 4: Compare Solution with Given Choices
Using the solution obtained, compare it with the given possible solutions:
1. [tex]\((8.69, -4.02)\)[/tex]
2. [tex]\((5.87, -3.45)\)[/tex]
3. [tex]\((-4.05, 1.21)\)[/tex]
4. [tex]\((-7.76, -1.66)\)[/tex]
The correct solution must match closely with the calculated values. Here, we see that the closest pair to our computed values is:
[tex]\[
(8.69, -4.02)
\][/tex]
### Conclusion
Therefore, the correct solution to the system of equations from the given choices is:
[tex]\[
\boxed{(8.69, -4.02)}
\][/tex]
