Given the system of inequalities:
[tex]\[
\begin{array}{l}
4x - 5y \leq 1 \\
\frac{1}{2}y - x \leq 3
\end{array}
\][/tex]

Which shows the given inequalities in slope-intercept form?

A. [tex]\[
\begin{array}{l}
y \leq \frac{4}{5}x - \frac{1}{5} \\
y \leq 2x + 6
\end{array}
\][/tex]

B. [tex]\[
\begin{array}{l}
y \geq \frac{4}{5}x - \frac{1}{5} \\
y \leq 2x + 6
\end{array}
\][/tex]

C. [tex]\[
\begin{array}{l}
y \geq -\frac{4}{5}x + \frac{1}{5} \\
y \geq 2x + 6
\end{array}
\][/tex]



Answer :

To convert the given system of inequalities into slope-intercept form, we need to re-arrange each inequality such that it follows the format [tex]\( y \leq mx + b \)[/tex] or [tex]\( y \geq mx + b \)[/tex]. Let's go through each inequality step-by-step:

1. Convert [tex]\( 4x - 5y \leq 1 \)[/tex] into slope-intercept form:
- Start by isolating [tex]\( y \)[/tex]. Subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[ -5y \leq -4x + 1 \][/tex]
- Next, divide every term by [tex]\( -5 \)[/tex]. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the direction of the inequality:
[tex]\[ y \geq \frac{4}{5}x - \frac{1}{5} \][/tex]

2. Convert [tex]\( \frac{1}{2}y - x \leq 3 \)[/tex] into slope-intercept form:
- First, eliminate the fraction by multiplying every term by 2:
[tex]\[ y - 2x \leq 6 \][/tex]
- Now, isolate [tex]\( y \)[/tex] by adding [tex]\( 2x \)[/tex] to both sides:
[tex]\[ y \leq 2x + 6 \][/tex]

The inequalities in slope-intercept form are:
[tex]\[ y \geq \frac{4}{5}x - \frac{1}{5} \][/tex]
and
[tex]\[ y \leq 2x + 6 \][/tex]

So, the correct choice is:

[tex]\[ y \geq \frac{4}{5}x - \frac{1}{5} \][/tex]
[tex]\[ y \leq 2x + 6 \][/tex]