Given the system of inequalities:
$4x - 5y \leq 1$
$\frac{1}{2}y - x \leq 3$
We are given a system of two linear inequalities:
1) \(4x - 5y \leq 1\)
2) \(\frac{1}{2}y - x \leq 3\)
Our goal is to analyze the region in the Cartesian coordinate plane defined by these inequalities. We will do this by rewriting both inequalities in the slope-intercept form (\(y \geq mx + b\) or \(y \leq mx + b\)).
For the first inequality \(4x - 5y \leq 1\):
Subtract \(4x\) from both sides:
\(-5y \leq -4x + 1\)
Divide by \(-5\) (remembering to reverse the inequality sign):
\(y \geq \frac{-4}{-5}x + \frac{1}{-5}\)
\(y \geq \frac{4}{5}x - \frac{1}{5}\)
This represents the set of points above or on the line \(y = \frac{4}{5}x - \frac{1}{5}\).
For the second inequality \(\frac{1}{2}y - x \leq 3\):
Add \(x\) to both sides:
\(\frac{1}{2}y \leq x + 3\)
Multiply by \(2\):
\(y \leq 2x + 6\)
This represents the set of points below or on the line \(y = 2x + 6\).
The boundary lines are \(L_1: y = \frac{4}{5}x - \frac{1}{5}\) and \(L_2: y = 2x + 6\).
Since their slopes are different (\(\frac{4}{5}
eq 2\)), the lines intersect. We find the point of intersection by setting the expressions equal:
\(\frac{4}{5}x - \frac{1}{5} = 2x + 6\)
Multiply by \(5\) to clear the denominators:
\(4x - 1 = 10x + 30\)
\(-6x = 31\)
\(x = -\frac{31}{6}\)
Substitute \(x\) into the second line equation:
\(y = 2(-\frac{31}{6}) + 6 = -\frac{31}{3} + \frac{18}{3} = -\frac{13}{3}\)
The lines intersect at \((-\frac{31}{6}, -\frac{13}{3})\).
The system defines an unbounded region in the coordinate plane. The region consists of all points \((x, y)\) such that:
\(\frac{4}{5}x - \frac{1}{5} \leq y \leq 2x + 6\)
The region is wedge-shaped, starting at the intersection point \((-\frac{31}{6}, -\frac{13}{3})\) and opening to the right, bounded below by the line \(y = \frac{4}{5}x - \frac{1}{5}\) and above by the line \(y = 2x + 6\). The final solution is the region defined by the system \(\boxed{\frac{4}{5}x - \frac{1}{5} \leq y \leq 2x + 6}\).
This type of analysis is common in linear programming, where one might try to maximize or minimize a profit or cost function given constraints on resources. For example, if \(x\) and \(y\) represent the number of two different types of products manufactured, these inequalities could represent constraints on labor hours and material usage. Finding the region of possible production values allows a manufacturer to determine their optimal operating point.