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? \\(\circ\\) \\(y \leq \frac{4}{5}x - \frac{1}{5}\\) \\(y \leq 2x + 6\\) \\(\circ\\) \\(y \geq \frac{4}{5}x - \frac{1}{5}\\) \\(y \leq 2x + 6\\) \\(\circ\\) \\(y \geq -\frac{4}{5}x + \frac{1}{5}\\) \\(y \geq 2x + 6\\)

Explanation:

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

We want to solve for \(y\). First, subtract \(4x\) from both sides:
\(-5y\leq - 4x + 1\)
When we divide both sides of an inequality by a negative number, the direction of the inequality sign changes. So divide both sides by \(- 5\):
\(y\geq\frac{4}{5}x-\frac{1}{5}\) (because dividing \(-4x\) by \(-5\) gives \(\frac{4}{5}x\), dividing \(1\) by \(-5\) gives \(-\frac{1}{5}\), and the inequality sign flips from \(\leq\) to \(\geq\))

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

First, add \(x\) to both sides:
\(\frac{1}{2}y\leq x + 3\)
Then multiply both sides by \(2\) to solve for \(y\):
\(y\leq2x + 6\) (multiplying \(x\) by \(2\) gives \(2x\), multiplying \(3\) by \(2\) gives \(6\), and the inequality sign remains \(\leq\) since we multiplied by a positive number)

Answer:

\(y\geq\frac{4}{5}x-\frac{1}{5}\)
\(y\leq2x + 6\) (the second option in the given choices)