Question

given the system of inequalities:
$4x - 5y \leq 1$
$\frac{1}{2}y - x \leq 3$
which shows the given inequalities in slope - intercept form?
$y \geq -\frac{4}{5}x + \frac{1}{5}$
$y \geq 2x + 6$
$y \leq \frac{4}{5}x - \frac{1}{5}$
$y \leq 2x + 6$
$y \geq \frac{4}{5}x - \frac{1}{5}$
$y \leq 2x + 6$

Explanation:

Step1: Convert \(4x - 5y \leq 1\) to slope - intercept form (\(y = mx + b\) form)

We start with the inequality \(4x - 5y \leq 1\). First, we want to isolate \(y\). Subtract \(4x\) from both sides: \(-5y\leq - 4x + 1\). Then, divide each term by \(-5\). When we divide an inequality by a negative number, the direction of the inequality sign changes. So we have \(y\geq\frac{4}{5}x-\frac{1}{5}\) (since \(\frac{-4x}{-5}=\frac{4}{5}x\) and \(\frac{1}{-5}=-\frac{1}{5}\)).

Step2: Convert \(\frac{1}{2}y - x\leq3\) to slope - intercept form

Start with \(\frac{1}{2}y - x\leq3\). Add \(x\) to both sides: \(\frac{1}{2}y\leq x + 3\). Then, multiply each side by \(2\) to solve for \(y\): \(y\leq2x + 6\) (because \(\frac{1}{2}y\times2=y\), \(x\times2 = 2x\) and \(3\times2=6\)).

Answer:

The third option (the one with \(y\geq\frac{4}{5}x-\frac{1}{5}\) and \(y\leq2x + 6\))