Identifying the Corner Point

Oct 13, 2025 by windailys.com 131 views

Introduction

In linear programming, the objective function is a mathematical expression that represents the goal of the problem. It is often represented as a linear equation in the form of $C = ax + by$ , where $C$ is the cost, $x$ and $y$ are the variables, and $a$ and $b$ are the coefficients. In this article, we will graph the constant-cost lines for the objective function $C = 8x + 5y$ through the points $(7,7)$ and $(12,12)$ . We will then use a straightedge to identify the corner point where the minimum cost occurs.

Graphing the Constant-Cost Lines

To graph the constant-cost lines, we need to find the equations of the lines that pass through the given points. The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)

Using this formula, we can find the equations of the lines passing through the points $(7,7)$ and $(12,12)$ .

Line 1: Passing through $(7,7)$

Let's find the equation of the line passing through the point $(7,7)$ . We can use the formula above with $(x_1, y_1) = (7, 7)$ and $(x_2, y_2) = (12, 12)$ .

y - 7 = \frac{12 - 7}{12 - 7}(x - 7)

Simplifying the equation, we get:

y - 7 = \frac{5}{5}(x - 7)

y - 7 = x - 7

y = x

This is the equation of the line passing through the point $(7,7)$ .

Line 2: Passing through $(12,12)$

Now, let's find the equation of the line passing through the point $(12,12)$ . We can use the formula above with $(x_1, y_1) = (7, 7)$ and $(x_2, y_2) = (12, 12)$ .

y - 7 = \frac{12 - 7}{12 - 7}(x - 7)

Simplifying the equation, we get:

y - 7 = \frac{5}{5}(x - 7)

y - 7 = x - 7

y = x

This is the equation of the line passing through the point $(12,12)$ .

Graphing the Constant-Cost Lines

Now that we have the equations of the lines passing through the points $(7,7)$ and $(12,12)$ , we can graph the constant-cost lines. The constant-cost lines are the lines that have the same slope as the objective function.

The objective function is $C = 8x + 5y$ . The slope of this line is $\frac{5}{8}$ .

The constant-cost lines have the same slope as the objective function, so they are also lines with a slope of $\frac{5}{8}$ .

To graph the constant-cost lines, we can use the equations of the lines passing through the points $(7,7)$ and $(12,12)$ .

Graphing the Constant-Cost Lines

The graph of the constant-cost lines is shown below:

	x	y
1	7	7
2	12	12

The graph shows that the constant-cost lines intersect at the point $(7,7)$ .

Identifying the Corner Point

The corner point is the point where the constant-cost lines intersect. In this case, the corner point is the point $(7,7)$ .

To confirm that this is the minimum cost point, we can use the objective function $C = 8x + 5y$ .

Substituting the values of $x$ and $y$ into the objective function, we get:

C = 8(7) + 5(7)

C = 56 + 35

C = 91

This is the minimum cost.

Conclusion

In this article, we graphed the constant-cost lines for the objective function $C = 8x + 5y$ through the points $(7,7)$ and $(12,12)$ . We then used a straightedge to identify the corner point where the minimum cost occurs. We confirmed our answer by substituting the values of $x$ and $y$ into the objective function.

The corner point is the point $(7,7)$ , and the minimum cost is $91$ .

References

[1] Linear Programming, by Bradley, Hax, and Magnanti
[2] Graphing Linear Equations, by Math Open Reference

Keywords

Graphing constant-cost lines
Objective function
Corner point
Minimum cost
Linear programming
Graphing linear equations