your constraint functions are:
6x + 9y <= 300
5x + 4y <= 180
x >= 0
y >= 0
all constraints have to be in minutes.
5 hours = 300 minutes
3 hours = 180 minutes
it helps to set up a table like the one below that allows you to visualize what's happening.
make sure that all constraints are dealing with the same units.
constraint units are in minutes for this problem.
product A product B
number of units x y
profit 6 7 maximize
machine 1 6 9 <= 300
machine 2 5 4 <= 180
your maximum profit will be when 20 units of product A and 20 units of product B are produced.
the maximum profit is 260 dollars.
you can solve this by using a linear optimization program such as can be found at https://www.zweigmedia.com/RealWorld/simplex.html, or by using the solver that comes with excel (it's an add in on the version of excel that i'm using), or by graphing.
i did all three and got the same answer, so it looks to be good, assuming i set the problem up correctly.
the linear optimization simplex solution is shown below.
the excel solution is shown below:
the graphical solution is shown below:
the desmos.com graphing software makes solving these types of problems easy.
using this software, you would graph the opposite of the inequalities.
the area of the graph that is not shaded is the region of feasibility.
the corner points of the region of feasibility are where the value of the objective function are evaluated.
for example, at the coordinate point of (20,20), the objective function of 6x + 7y is evaluated to get 6*20 + 7*20 = 260.
all the constraint functions have to be satisfied as well.
x and y are greater than or equal to 0.
6x + 9y = 6*20 + 9*20 = 300 which is less than or equal to 300.
5x + 4y = 5*20 + 4*20 = 180 which is less than or equal to 180.