Question:
For the feasible region shown below, the non-trivial constraints of the linear programming problem are
For the feasible region shown below, the non-trivial constraints of the linear programming problem are
Show Hint
Use the intercept form \( \frac{x}{x\text{-int}} + \frac{y}{y\text{-int}} = 1 \) to find line equations instantly from graphs.
Check the origin \( (0, 0) \) in your derived inequality to verify the direction of the shaded region.
Check the origin \( (0, 0) \) in your derived inequality to verify the direction of the shaded region.
Updated On: Sep 10, 2026
- \( x + y \leq 5, x + 3y \leq 9 \)
- \( x + y \leq 5, x + 3y \geq 9 \)
- \( x + y \geq 5, x + 3y \leq 9 \)
- \( x + y \geq 5, 3x + y \leq 9 \)
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The Correct Option is A
Solution and Explanation
Concept:
• The equation of a line passing through \( (a, 0) \) and \( (0, b) \) is \( \frac{x}{a} + \frac{y}{b} = 1 \).
• If the shaded region includes the origin \( (0, 0) \), the inequality for a line \( ax + by \leq c \) (where \( c > 0 \)) is generally \( \leq \).
Step 1: Find the equation of the first boundary line
Looking at the graph, one line passes through \( (5, 0) \) and \( (0, 5) \).
Using the intercept form \( \frac{x}{5} + \frac{y}{5} = 1 \):
\[ x + y = 5 \]
Since the region is shaded below the line (towards the origin), the inequality is:
\[ x + y \leq 5 \]
Step 2: Find the equation of the second boundary line
The second line passes through \( (9, 0) \) and \( (0, 3) \).
Using the intercept form \( \frac{x}{9} + \frac{y}{3} = 1 \):
Multiplying by 9 on both sides:
\[ x + 3y = 9 \]
Since the region is shaded below this line as well, the inequality is:
\[ x + 3y \leq 9 \]
Step 3: Identify non-trivial constraints
Non-trivial constraints exclude the standard non-negativity constraints (\( x \geq 0, y \geq 0 \)).
Thus, the constraints are \( x + y \leq 5 \) and \( x + 3y \leq 9 \).
• The equation of a line passing through \( (a, 0) \) and \( (0, b) \) is \( \frac{x}{a} + \frac{y}{b} = 1 \).
• If the shaded region includes the origin \( (0, 0) \), the inequality for a line \( ax + by \leq c \) (where \( c > 0 \)) is generally \( \leq \).
Step 1: Find the equation of the first boundary line
Looking at the graph, one line passes through \( (5, 0) \) and \( (0, 5) \).
Using the intercept form \( \frac{x}{5} + \frac{y}{5} = 1 \):
\[ x + y = 5 \]
Since the region is shaded below the line (towards the origin), the inequality is:
\[ x + y \leq 5 \]
Step 2: Find the equation of the second boundary line
The second line passes through \( (9, 0) \) and \( (0, 3) \).
Using the intercept form \( \frac{x}{9} + \frac{y}{3} = 1 \):
Multiplying by 9 on both sides:
\[ x + 3y = 9 \]
Since the region is shaded below this line as well, the inequality is:
\[ x + 3y \leq 9 \]
Step 3: Identify non-trivial constraints
Non-trivial constraints exclude the standard non-negativity constraints (\( x \geq 0, y \geq 0 \)).
Thus, the constraints are \( x + y \leq 5 \) and \( x + 3y \leq 9 \).
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