Questions: Graph the solution of the system of linear inequalities. [ 5 x-3 y leq 15 y leq-5 ]

Transcript text: Graph the solution of the system of linear inequalities. \[ \left\{\begin{array}{l} 5 x-3 y \leq 15 \\ y \leq-5 \end{array}\right. \]

Solution

Solution Steps

Step 1: Solve the first inequality

The first inequality is: \[ 5x - 3y \leq 15 \]

To find the boundary line, we set the inequality to equality: \[ 5x - 3y = 15 \]

Solving for $ y $: \[ y = \frac{5}{3}x - 5 \]

Step 2: Solve the second inequality

The second inequality is: \[ y \leq -5 \]

This is a horizontal line at $ y = -5 $.

Step 3: Determine the solution region

The solution region is the area where both inequalities are satisfied. This is the region below both lines:

Below the line $ y = \frac{5}{3}x - 5 $
Below the line $ y = -5 $

Final Answer

The solution to the system of inequalities is the region below both lines.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (5/3)x - 5", "y = -5"], "latex_expressions": ["$y = \\frac{5}{3}x - 5$", "$y = -5$"]}

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