Consider the following system of equations.
[tex]\[
\begin{aligned}
-2x + 5y & = 19 \\
y & = -\frac{5}{6}x - \frac{1}{6}
\end{aligned}
\][/tex]

Use this graph of the system to approximate its solution.

A. [tex]\(\left(-\frac{13}{4}, \frac{5}{2}\right)\)[/tex]

B. [tex]\(\left(\frac{5}{2}, -\frac{13}{4}\right)\)[/tex]

C. [tex]\(\left(-\frac{5}{2}, \frac{13}{4}\right)\)[/tex]

D. [tex]\(\left(\frac{13}{4}, -\frac{5}{2}\right)\)[/tex]



Answer :

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]