Question 1067023
it looks like you want a solution that uses exactly the number of hours of machine time and labor time available.


a small chair takes 30 minutes of machine time and 60 minutes of labor.
a large chair takes 45 minutes of machine time and 80 minutes of labor.
you have a total of 34 hours of machine time available.
you have a total of 64 hours of labor time available.


let x = number of small chairs.
let y = number of large chairs.


your constraint equations are:


machine time:
30x + 45y = 34*60


labor time:
60x + 80y = 64*60


the total machine time and labor time are converted to minutes by multiplying them by 60.


the equations become:


30x + 45y = 2040
60x + 80y = 3840


solve these 2 equations simultaneously and you'll get the number of small and large chairs that can be built with those available hours.


multiply the first equation by 2 and leave the second equation as is to get:


60x + 90y = 4080
60x + 80y = 3840


subtract the second equation from the first to get:


10y = 240


solve for y to get y = 24


since 30x + 45y = 2040, and since y = 24, then replace y with 24 and you get:
30x + 45*24 = 2040
simplify to get 30x + 1080 = 2040
subtract 1080 from both sides of the equation to get:
30x = 960
solve for x to get x = 32


your solution is that x = 32 and y = 24.


that means 32 small chairs and 24 large chairs can be built based on the constraints.