finding corner points of feasible region using ti84 calculator

Interactive Feasible Region Corner Point Finder

This tool simulates finding corner points for a linear programming problem with two variables (x, y) and up to three constraints, plus non-negativity (x ≥ 0, y ≥ 0).

What is Finding Corner Points of a Feasible Region?

In linear programming, a “feasible region” is the set of all possible points that satisfy a system of linear inequalities (called constraints). These points represent all the valid solutions to the problem. The corner points (or vertices) of this region are critical because of the Corner Point Theorem, which states that the optimal solution (maximum or minimum value of the objective function) will always occur at one of these corners. Therefore, finding corner points is a fundamental step in solving optimization problems.

This process is frequently taught using graphing calculators like the TI-84, which can visualize the inequalities and find their intersection points. A Linear Programming Calculator simplifies this by automating the algebraic solutions.

The “Formula”: How Corner Points are Found

There isn’t a single formula for finding corner points, but rather an algebraic method. A corner point is the intersection of two or more boundary lines of the feasible region. For a 2-variable problem, you find the corners by taking the equations of the boundary lines two at a time and solving the resulting system of two linear equations.

For a system:

A₁x + B₁y = C₁

A₂x + B₂y = C₂

The intersection (x, y) can be found using methods like substitution or elimination. Once an intersection is found, it must be tested against all constraints to ensure it lies within the feasible region.

Variables in Constraint Equations

Variable	Meaning	Unit	Typical Range
x, y	Decision Variables	Unitless or context-dependent (e.g., items, hours)	Non-negative (≥ 0)
A, B	Coefficients	Rate per unit of decision variable	Any real number
C	Constraint Limit	Total resource available	Usually a positive number

Practical Examples

Example 1: Manufacturing Scenario

A company produces two products, X and Y. Product X requires 2 hours of labor and 1 unit of material. Product Y requires 1 hour of labor and 1 unit of material. They have 100 hours of labor and 80 units of material available. The goal is to find the corner points of the production possibilities.

• Constraint 1 (Labor): 2x + 1y ≤ 100
• Constraint 2 (Material): 1x + 1y ≤ 80
• Non-negativity: x ≥ 0, y ≥ 0

Using the calculator above with these inputs, the resulting corner points are: (0, 0), (50, 0), (0, 80), and (20, 60). These represent the vertices of the feasible production combinations.

Example 2: Diet Planning

A person needs at least 40mg of protein and 5mg of iron. Food A has 10mg protein and 2mg iron per serving. Food B has 20mg protein and 1mg iron per serving.

• Constraint 1 (Protein): 10x + 20y ≥ 40
• Constraint 2 (Iron): 2x + 1y ≥ 5

This demonstrates a minimization problem with “greater than or equal to” constraints, leading to an unbounded feasible region with different corner points. A tool to graph inequalities is essential here.

How to Use This Calculator and a TI-84

Using This Online Calculator:

1. Enter Coefficients: For each constraint of the form Ax + By ≤ C, type the values for A, B, and C into the corresponding input fields.
2. View Results: The calculator automatically solves for the intersection of all boundary lines (including the x and y axes for non-negativity).
3. Check Feasibility: It then tests each intersection point against all constraints.
4. Analyze Output: The final list shows only the valid corner points that form the vertices of the feasible region. The chart provides a visual confirmation.

Steps for a TI-84 Calculator:

1. Solve for Y: The TI-84 requires you to enter equations in “Y=” form. You must first isolate ‘y’ in each inequality. For example, `2x + y ≤ 100` becomes `y ≤ 100 – 2x`.
2. Enter Equations: Press the `[Y=]` button. Enter each solved inequality into Y1, Y2, etc.
3. Set Shading: Move the cursor to the far left of the Y= line. Press `[ENTER]` repeatedly to change the graph style. For a ‘≤’ inequality, choose the “shade below” icon (a triangle pointing down). For ‘≥’, choose “shade above”.
4. Graph: Press `[GRAPH]`. You will see the overlapping shaded areas, which form the feasible region.
5. Find Intersections: To find a corner point where two lines cross, use the `[2nd]` -> `[TRACE]` (CALC) menu and select `5: intersect`. The calculator will prompt you to select the “First curve” and “Second curve” (the two lines you want to find the intersection of). Press `[ENTER]` for each and then `[ENTER]` on the “Guess?”. The coordinates of the intersection will be displayed.
6. Find Axis Intercepts: To find where a line crosses an axis, you can use the `[2nd]` -> `[TRACE]` (CALC) menu and select `1: value`. Enter `x=0` to find the y-intercept. To find the x-intercept, you can use `2: zero`, but it’s often easier to set Y2=0 and find the intersection with your first line.

Key Factors That Affect the Feasible Region

• Constraint Limits (C values): Increasing or decreasing the ‘C’ value shifts the boundary line, which can expand, shrink, or move the feasible region.
• Coefficients (A and B values): Changing the coefficients alters the slope of the boundary line, which pivots the line and changes the shape of the feasible region.
• Inequality Direction (≤ or ≥): Flipping the inequality sign reverses the shaded area for that constraint, which can dramatically change or even eliminate the feasible region.
• Number of Constraints: Adding more constraints can only shrink or leave the feasible region unchanged. It can never expand it. A system of equations solver can help manage many constraints.
• Non-Negativity Constraints (x≥0, y≥0): These constraints typically confine the feasible region to the first quadrant of the graph, which is standard for most real-world problems where quantities can’t be negative.
• Parallel Lines: If two constraint lines are parallel, they may not create a corner point, or they might form one entire edge of the feasible region.

Frequently Asked Questions (FAQ)

What does a corner point represent in the real world?

It represents a point of maximum resource utilization. For example, in a production problem, a corner point is a production plan that uses up all of one or more available resources (like labor hours or materials).

What is an unbounded feasible region?

An unbounded region occurs when the feasible area is not fully enclosed. It continues infinitely in one or more directions. This is common in minimization problems. Such regions will still have corner points, but a maximum value for the objective function may not exist.

Why did my intersection point not show up as a corner?

Because it was not feasible. An intersection of two lines is only a corner point if it also satisfies all *other* constraints in the system. Our calculator automatically filters out these infeasible points.

How do I handle a constraint like ‘x ≥ 10’ on the TI-84?

The TI-84’s “Y=” editor is for functions of x. For vertical lines like x = 10, you can’t enter it directly. However, the TI-84’s “Inequalz” application allows for vertical line constraints. Without the app, you must visually account for that boundary on the graph.

Can a feasible region be empty?

Yes. If the constraints are contradictory (for example, x ≤ 5 and x ≥ 10), there is no point that can satisfy both. In this case, there is no feasible region and therefore no solution.

Does the optimal solution have to be a corner point?

Yes, according to the Corner Point Theorem. If an optimal solution exists, it will be at one of the corner points. In the rare case that the objective function line is parallel to one of the edges of the feasible region, there can be multiple optimal solutions along that entire edge, which includes two corner points.

How does this relate to an objective function?

Finding the corner points is the first major step. The next step is to plug the (x, y) coordinates of each corner point into the objective function (e.g., Profit = 5x + 7y) to see which point yields the highest or lowest value.

What if the TI-84 gives an “ERR: NO SIGN CHNG” when finding an intersection?

This error means the calculator could not find an intersection within its search bounds. This usually happens if the lines do not intersect or if your viewing window is not set correctly to see the intersection point.