Answered

Given the system of inequalities:

$\begin{array}{l}

4 x-5 y \leq 1 \\

\frac{1}{2} y-x \leq 3

\end{array}$



Which shows the given inequalities in slope-intercept form?


A

$\begin{array}{l}

y \leq \frac{4}{5} x-\frac{1}{5} \\

y \leq 2 x+6

\end{array}$


B

$\begin{array}{l}

y \geq \frac{4}{5} x-\frac{1}{5} \\

y \leq 2 x+6

\end{array}$


C

$\begin{array}{l}

y \geq-\frac{4}{5} x+\frac{1}{5} \\

y \geq 2 x+6

\end{array}$

Verified Answer :

  • Isolate \(y\) in the first inequality by subtracting \(4x\) and dividing by \(-5\), which reverses the sign to obtain \(y \geq \frac{4}{5}x - \frac{1}{5}\).
  • Isolate \(y\) in the second inequality by adding \(x\) and multiplying by 2 to obtain \(y \leq 2x + 6\).
  • Match the derived forms with the given options to identify the correct choice.
  • Final answer: \(\boxed{\begin{array}{l} y \geq \frac{4}{5}x - \frac{1}{5} \ y \leq 2x + 6 \end{array}}\)

Explanation

  1. Analysis of the Problem

The given system of linear inequalities is:
1) \(4x - 5y \leq 1\)
2) \(\frac{1}{2}y - x \leq 3\)

Our goal is to rewrite both inequalities in slope-intercept form, which is typically of the form \(y \geq mx + b\) or \(y \leq mx + b\), where \(m\) represents the slope and \(b\) is the \(y\)-intercept. This allows us to easily visualize the regions defined by these inequalities on a coordinate plane.

  1. Rewriting the First Inequality

To rewrite the first inequality, \(4x - 5y \leq 1\), we isolate the \(y\) term:
Subtract \(4x\) from both sides:
\(-5y \leq -4x + 1\)

Now, divide both sides by \(-5\). Remember that when dividing an inequality by a negative number, the direction of the inequality sign must be reversed:
\(y \geq \frac{-4x}{-5} + \frac{1}{-5}\)
\(y \geq \frac{4}{5}x - \frac{1}{5}\)

  1. Rewriting the Second Inequality

To rewrite the second inequality, \(\frac{1}{2}y - x \leq 3\), we isolate the \(y\) term:
Add \(x\) to both sides:
\(\frac{1}{2}y \leq x + 3\)

Now, multiply both sides by 2. Since 2 is a positive number, the direction of the inequality sign remains the same:
\(y \leq 2(x + 3)\)
\(y \leq 2x + 6\)

  1. Conclusion and Final Answer

Comparing our results to the given options, we have:
1) \(y \geq \frac{4}{5}x - \frac{1}{5}\)
2) \(y \leq 2x + 6\)

This matches the option:
\[ \begin{array}{l} y \geq \frac{4}{5}x - \frac{1}{5} \ y \leq 2x + 6 \end{array} \]

Examples

In real-life scenarios, such as production planning, linear inequalities help define feasible regions. Imagine a factory that produces two types of goods, \(x\) and \(y\), subject to resource constraints represented by inequalities. By converting these to slope-intercept form, the factory manager can graph these constraints on a coordinate plane to visualize the 'feasible region'—the set of all possible production quantities that satisfy all constraints simultaneously, ensuring both efficiency and regulatory compliance.

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