Question 823732: Maximize p = 5x + 3y subject to
2x + 3y <= 12
3x + y >= 16
x + y >= 3
2x + y >= 6 Answer by KMST(5328) (Show Source): You can put this solution on YOUR website! represents a line on the x-y plane that divides the at plane into two halves. represents the half of the x-y plane that contains (0,0), the origin,
because with , .
Graphing the line for is easy.
We just need 2 points, and we can find the x- and y-intercepts very easily: -->-->-->-->
gives us the y-intercept at (0,4). -->-->-->-->
gives us the y-intercept at (0,4).
We can graph that line, which is part of the solution to ,
and we can indicate with a little arrow which half of the plane is the whole solution:
We can do the same for the inequalities , , and
We can easily see that (0,0) the origin is not part of the solution to any of those 3 inequalities,
We can easily find the intercepts and graph the boundary lines for , which has intercepts at (0,3) and (3,0) , and , which has intercepts at (0,6) and (3,0) .
The line with intercepts at (0,16) and (16/3,0) looks a little more cumbersome for graphing, but transforming the equation, <--> ,
makes it easier to find points more amenable to graphing, such as
(6,-2) from --> and
(3,7) from --> .
For each inequality, the solution is the boundary line plus side of the boundary line that does not contain the origin.
We can add to the graph the 3 lines above, with little arrows showing which half of the plane is the solution to the inequality
The graph of the 4 inequalities looks like this: , or better yet
The points that satisfy all 3 inequalities form that quadrilateral in the middle of the last graph.
It would be good to know the coordinates of its vertices.
We know that (3,0) is one of them because it was the x-intercept for the red and green lines.
Point (1.5,3) seems to be the intersection of the black and green lines.
Substituting the coordinates into and , we find that they satisfy both equations, so (1.5,3) is the intersection of the black and green lines. (That was solving ta system of linear equations by graphing.
The other two vertices are not so easy. -->-->-->-->-->-->
gives us the intersection of the black and blue lines. -->-->-->-->-->-->--> gives us the intersection of the black and blue lines.
Now we have to find at which vertex or vertices the function has the greatest value.
The maximum happens at or and that maximum is which is approximately .