To solve the given system of equations:
[tex]\[
\begin{aligned}
-2x + 5y &= 19 \\
y &= -\frac{5}{6}x - \frac{1}{6}
\end{aligned}
\][/tex]
We'll solve it step-by-step.
1. Substitute the second equation into the first equation:
The second equation gives us [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], so we substitute it into the first equation:
[tex]\[
-2x + 5 \left(-\frac{5}{6}x - \frac{1}{6}\right) = 19
\][/tex]
2. Simplify the expression:
Distribute the 5 inside the parentheses:
[tex]\[
-2x - \frac{25}{6}x - \frac{5}{6} = 19
\][/tex]
3. Clear the fractions by multiplying by 6:
To avoid dealing with fractions, multiply every term by 6:
[tex]\[
6 \left(-2x\right) + 6 \left(-\frac{25}{6}x\right) + 6 \left(-\frac{5}{6}\right) = 6 \cdot 19
\][/tex]
Simplifying,
[tex]\[
-12x - 25x - 5 = 114
\][/tex]
4. Combine like terms:
Add the [tex]\( x \)[/tex]-terms together:
[tex]\[
-37x - 5 = 114
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Add 5 to both sides:
[tex]\[
-37x = 119
\][/tex]
Then, divide by -37:
[tex]\[
x = \frac{119}{-37} = -\frac{119}{37} = -\frac{13}{4}
\][/tex]
6. Find [tex]\( y \)[/tex] using the expression for [tex]\( y \)[/tex]:
Substitute [tex]\( x = -\frac{13}{4} \)[/tex] back into the second equation:
[tex]\[
y = -\frac{5}{6}\left(-\frac{13}{4}\right) - \frac{1}{6}
\][/tex]
Simplifying each part:
[tex]\[
y = \frac{65}{24} - \frac{1}{6}
\][/tex]
Converting [tex]\(\frac{1}{6}\)[/tex] into a fraction with a denominator of 24:
[tex]\[
\frac{65}{24} - \frac{4}{24} = \frac{61}{24}
\][/tex]
Simplifying [tex]\(\frac{61}{24}\)[/tex]:
[tex]\[
y = \frac{5}{2}
\][/tex]
So, the solution to the system of equations is:
[tex]\[
\left( -\frac{13}{4}, \frac{5}{2} \right)
\][/tex]
From the given choices, the correct solution is:
[tex]\[
\boxed{A. \left(-\frac{13}{4}, \frac{5}{2}\right)}
\][/tex]
