System of inequalities

[thepapercup]

Hey again guys. =p
I is back for some more math help.

Here is the question.

x>(or equal to) 0
y>(or equal to) 0
x+y<(or equal to) 8
2x+3y<(or equal to) 21

I need to solve the last line for x or y, but I can't really remember how exactly to do that.
From there I know what to do.

Thanks for the help.

 

[soroban]

Hello, thepapercup!

\(\displaystyle \begin{array}{c}x\:\geq \:0 \\ y \:\geq\:0 \\ x + y \:\leq \:8 \\ 2x + 3y \:\leq\:21 \end{array}\)

I need to solve the last line for x or y. . . . . why?

Graph the lines and shade the appropriate regions.

\(\displaystyle x \:=\:0 \text{ is the y-axis.}\)
. . \(\displaystyle \text{It says }\geq\text{ . . . shade the region to the }right\text{ of the line.}\)

\(\displaystyle y \:=\:0 \text{ is the x-axis.}\)
. . \(\displaystyle \text{It says }\geq\text{ . . . shade the region }above\text{ the line.}\)

\(\displaystyle \text{We are already limited to Quadrant 1.}\)


\(\displaystyle \text{The line }x + y \:=\:8\text{ has intercepts }(8,0)\text{ and }(0,8)\)
\(\displaystyle \text{Plot the points and draw the line.}\)
. . \(\displaystyle \text{It says }\leq \text{ . . . shade the region }below\text{ the line.}\)

\(\displaystyle \text{The line }2x + 3y \:\leq \:21\text{ has intercepts }(10.5,\:0)\text{ and }(0,7)\)
\(\displaystyle \text{Plot the points and draw the line.}\)
. . \(\displaystyle \text{It says }\leq\text{ . . . shade the region }below\text{ the line.}\)


\(\displaystyle \text{The final region looks like this:}\)

        |
      8 *
        | *
        |   *
        |     *
      7 o..     *
        |:::::*.. *
        |:::::::::..o
        |::::::::::::.*   *
    - - o - - - - - - - o - - - * -  -
        |               8      10.5

The shaded region represents all the points that satisfy the four inequalities.
This includes the point on the lines, too.

\(\displaystyle \text{We can find the coordinates of that fourth vertex: }\:\begin{array}{ccc}x + y &=& 8 \\ 2x + 3y &=& 21 \end{array}\)
\(\displaystyle \text{Solve the system and get: }\;x = 3,\;y = 5\)