5.1 Q-4

Question Statement

Graph the solution region of the following systems of linear inequalities and find the corner points for each case:

1. $2x - 3y \leq 6$, $2x + 3y \leq 12$, $x \geq 0$
2. $x + y \leq 5$, $2 - 2x + y \leq 2$, $y \geq 0$
3. $3x + 7y \leq 21$, $2x - y \leq -3$, $y \geq 0$
4. $3x + 2y \geq 6$, $2x + 3y \leq 6$, $y \geq 0$
5. $5x + 7y \leq 35$, $-x + 3y \leq 3$, $x \geq 0$

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Background and Explanation

To solve these systems of inequalities, we:

• Graph each inequality by converting them into linear equations (i.e., replacing $\leq$ or $\geq$ with $=$).
• Identify key points where the lines intersect.
• The solution region is the area where the shaded regions of all inequalities overlap.
• The corner points are the intersection points of these boundary lines.

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Solution

Case (i)

Inequalities:

$$2x - 3y \leq 6, \quad 2x + 3y \leq 12, \quad x \geq 0$$

1. Convert inequalities to equations:

   • $2x - 3y = 6$
   • $2x + 3y = 12$

2. Find intersection points:

   • For $2x - 3y = 6$, set $x = 0$ to find $y = -2$. So, the point $(0, -2)$ is on the line.
   • Set $y = 0$, then $x = 3$. So, the point $(3, 0)$ is on the line. Similarly, solve for the other equation:
   • For $2x + 3y = 12$, set $x = 0$ to find $y = 4$. So, the point $(0, 4)$ is on the line.
   • Set $y = 0$, then $x = 6$. So, the point $(6, 0)$ is on the line.

3. Find the corner points where the two lines intersect:

   • Solving the system $2x - 3y = 6$ and $2x + 3y = 12$:
     • $x = \frac{9}{2}, y = 1$.
   • So, the corner points are: $\left(\frac{9}{2}, 1\right), (0, 4), (0, -2)$.

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Case (ii)

Inequalities:

$$x + y \leq 5, \quad 2 - 2x + y \leq 2, \quad y \geq 0$$

1. Convert inequalities to equations:

   • $x + y = 5$
   • $-2x + y = 2$

2. Find intersection points:

   • For $x + y = 5$, set $x = 0$ to find $y = 5$. So, the point $(0, 5)$ is on the line.
   • Set $y = 0$, then $x = 5$. So, the point $(5, 0)$ is on the line.

   Similarly, solve for the other equation:

   • For $-2x + y = 2$, set $x = 0$ to find $y = 2$. So, the point $(0, 2)$ is on the line.
   • Set $y = 0$, then $x = -1$. So, the point $(-1, 0)$ is on the line.

3. The corner points are: $(0, 5), (5, 0), (0, 2), (-1, 0)$.

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Case (iii)

Inequalities:

$$3x + 7y \leq 21, \quad 2x - y \leq -3, \quad y \geq 0$$

1. Convert inequalities to equations:

   • $3x + 7y = 21$
   • $2x - y = -3$

2. Find intersection points:

   • For $3x + 7y = 21$, set $x = 0$ to find $y = 3$. So, the point $(0, 3)$ is on the line.
   • Set $y = 0$, then $x = 7$. So, the point $(7, 0)$ is on the line.

   Similarly, solve for the other equation:

   • For $2x - y = -3$, set $x = 0$ to find $y = 3$. So, the point $(0, 3)$ is on the line.
   • Set $y = 0$, then $x = -1.5$. So, the point $(-1.5, 0)$ is on the line.

3. The corner points are: $(0, 3), (7, 0), (-1.5, 0)$.

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Case (iv)

Inequalities:

$$3x + 2y \geq 6, \quad 2x + 3y \leq 6, \quad y \geq 0$$

1. Convert inequalities to equations:

   • $3x + 2y = 6$
   • $2x + 3y = 6$

2. Find intersection points:

   • For $3x + 2y = 6$, set $x = 0$ to find $y = 3$. So, the point $(0, 3)$ is on the line.
   • Set $y = 0$, then $x = 2$. So, the point $(2, 0)$ is on the line.

   Similarly, solve for the other equation:

   • For $2x + 3y = 6$, set $x = 0$ to find $y = 2$. So, the point $(0, 2)$ is on the line.
   • Set $y = 0$, then $x = 6$. So, the point $(6, 0)$ is on the line.

3. The corner points are: $(0, 3), (2, 0), (6, 0)$.

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Case (v)

Inequalities:

$$5x + 7y \leq 35, \quad -x + 3y \leq 3, \quad x \geq 0$$

1. Convert inequalities to equations:

   • $5x + 7y = 35$
   • $-x + 3y = 3$

2. Find intersection points:

   • For $5x + 7y = 35$, set $x = 0$ to find $y = 5$. So, the point $(0, 5)$ is on the line.
   • Set $y = 0$, then $x = 7$. So, the point $(7, 0)$ is on the line.

   Similarly, solve for the other equation:

   • For $-x + 3y = 3$, set $x = 0$ to find $y = 1$. So, the point $(0, 1)$ is on the line.
   • Set $y = 0$, then $x = -3$. So, the point $(-3, 0)$ is on the line.

3. The corner points are: $(0, 5), (7, 0), (0, 1), (-3, 0)$.

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Key Formulas or Methods Used

• Graphing Linear Inequalities:
  Convert inequalities to equations to graph the boundary lines and find points of intersection.

• Solving Systems of Equations:
  Use substitution or elimination methods to find the intersection points.

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Summary of Steps

1. Convert each inequality to an equation.
2. Graph the boundary lines by plotting key points.
3. Solve the system of equations to find the intersection points.
4. Identify the corner points of the solution region.
5. Verify the corner points by checking if they satisfy all inequalities.

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