Labelling variables on a graph

[abcd]

The problem:

A company has 120 man-hours of labor available per week. A table takes 15 and a chair takes 7. The warehouse stores 200 board feet of wood, and is restocked weekly. A table takes 32 board feet of materials to build, and a chair takes 8.

This situation is represented by a graph: http://i45.tinypic.com/11maixz.jpg

I have to figure out what each variable on the graph (A, B, and C) represents.

What I've done so far:

So far I've figured out that this situation can be represented by 15x + 7y <= 120, 32x + 8y <= 200, x >= 0, y >= 0
x=tables, y=chairs

But the graph part is confusing me. Please help.

 

[mmm4444bot]

The symbols A, B, and C are not variables. (Variables are symbols that represent numbers.)

In this diagram, the symbols A, B, and C are just names for identifying three parts of the graph.

This graph shows a "picture" of the solutions to the set of four linear inequalities in your exercise.

A is shown as a line segment, but you can think of A as a line that continues forever in both directions.

B is shown as another line segment, but you can think of B as an entire line, too.

C is a shaded region of the xy-plane.

Do you know? The solution to a single inequality in x and y is a set of (x, y) pairs. Each solution can be interpreted as a point in the xy-plane. If you plot all of the solution points, you get a "region". We show this by shading.

So, the shaded region C is a "picture" of the solutions to the system of four inequalities that you posted.

15x + 7y <= 120 is a single inequality. The solutions are all of the (x, y) pairs of values that lie below the corresponding line.

Therefore, as a way to "display" all of the solutions to this single inequality, we graph the line 15x + 7y = 120 and shade everything below it.

The same goes for the other three inequalities.

Now, in order for some pair of values (x, y) to satisfy ALL FOUR inequalities at once, it must lie within the shaded region C.

In other words, region C is the area where all four solutions overlap.

You can determine which line belongs to which label (A or B) by solving each equation for y and comparing slopes. One slope is greater than the other, and that would be line B because it's steeper.

Check out the lesson on graphing a system of linear inequalities HERE. If you're still confused about how to graph inequalities, let us know.

Cheers ~ Mark

 

[abcd2]

So, A represents 32x+8y <= 200, or how many tables and chairs can be built with the amount of wood available, B represents 15x+7y<=120, or how many tables and chairs can be built per week with the number of hours of labor available, and C represents the combination of these two things?

 

[mmm4444bot]

Please tell me the slope of each of these lines:

32x + 8y = 200

15x + 7y = 120