To determine whether Hugo's solution to the given system of equations is correct, we will need to solve the system step-by-step and then compare it with Hugo’s solution. The provided system is:
[tex]\[
\begin{cases}
2x - 3y = -11 \\
3x + 2y = -5
\end{cases}
\][/tex]
Step 1: Let's use the method of elimination to solve the system. First, we need to align the coefficients for one of the variables (let’s choose [tex]\(y\)[/tex]) so that we can eliminate one variable.
Multiply the first equation by 2 and the second equation by 3:
[tex]\[
\begin{cases}
4x - 6y = -22 \quad \text{(Equation 1 multiplied by 2)} \\
9x + 6y = -15 \quad \text{(Equation 2 multiplied by 3)}
\end{cases}
\][/tex]
Step 2: Now, add both equations to eliminate [tex]\(y\)[/tex]:
[tex]\[
4x - 6y + 9x + 6y = -22 - 15
\][/tex]
This simplifies to:
[tex]\[
13x = -37
\][/tex]
Step 3: Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -\frac{37}{13}
\][/tex]
Step 4: Substitute [tex]\(x = -\frac{37}{13}\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]. We will use the first equation [tex]\(2x - 3y = -11\)[/tex]:
[tex]\[
2\left(-\frac{37}{13}\right) - 3y = -11
\][/tex]
Simplify:
[tex]\[
-\frac{74}{13} - 3y = -11
\][/tex]
Convert [tex]\(-11\)[/tex] to a fraction with the same denominator:
[tex]\[
-\frac{74}{13} - 3y = -\frac{143}{13}
\][/tex]
Isolate [tex]\(3y\)[/tex]:
[tex]\[
-3y = -\frac{143}{13} + \frac{74}{13}
\][/tex]
[tex]\[
-3y = -\frac{69}{13}
\][/tex]
[tex]\[
y = \frac{69}{39}
\][/tex]
[tex]\[
y = \frac{23}{13}
\][/tex]
So, the solution we obtain is:
[tex]\[
x = -\frac{37}{13}, \quad y = \frac{23}{13}
\][/tex]
Step 5: Compare this solution to Hugo's solution, which was [tex]\(x = -2\)[/tex] and [tex]\(y = \frac{1}{2}\)[/tex]. Clearly, our solutions do not match Hugo's.
Conclusion:
Hugo's solution is incorrect. The correct solution is [tex]\(x = -\frac{37}{13}\)[/tex] and [tex]\(y = \frac{23}{13}\)[/tex].
