as a junior entrepreneur, layla decided to offer to her schools graduating class a deal they could not…

as a junior entrepreneur, layla decided to offer to her schools graduating class a deal they could not refuse: graduation pictures with their favourite actor in the background. she offers the choice of a small picture for $5 or a large picture for $12. layla expects to sell at least 25 more small pictures than large pictures, and she expects to make less than $500. a) define the variables. b) determine if there are any restrictions on the variables. c) write a system of linear inequalities to represent this situation. d) graph the system of linear inequalities. e) use your graph to determine if layla could have sold 50 small and 20 large pictures.

as a junior entrepreneur, layla decided to offer to her schools graduating class a deal they could not refuse: graduation pictures with their favourite actor in the background. she offers the choice of a small picture for $5 or a large picture for $12. layla expects to sell at least 25 more small pictures than large pictures, and she expects to make less than $500. a) define the variables. b) determine if there are any restrictions on the variables. c) write a system of linear inequalities to represent this situation. d) graph the system of linear inequalities. e) use your graph to determine if layla could have sold 50 small and 20 large pictures.

Answer

Answer:

(a) Let (x) be the number of small pictures and (y) be the number of large pictures. (b) (x\geq0), (y\geq0), (x - y\geq25), (5x + 12y<500) (c) Yes, non - negative values for (x) and (y) (since you can't sell a negative number of pictures), and the inequality (x - y\geq25) and (5x + 12y<500) are restrictions. (d) To graph:

  • For (x - y=25), rewrite as (y=x - 25), has (y) - intercept ((0,- 25)) and (x) - intercept ((25,0)), draw a solid line (because (x - y\geq25)) and shade the region where (y\leq x - 25).
  • For (5x + 12y=500), rewrite as (y=-\frac{5}{12}x+\frac{125}{3}), has (y) - intercept ((0,\frac{125}{3})\approx41.67) and (x) - intercept ((100,0)), draw a dashed line (because (5x + 12y<500)) and shade the region below the line. Also, shade the first - quadrant since (x\geq0) and (y\geq0). (e) Substitute (x = 50) and (y = 20) into the inequalities:
  • For (x - y): (50-20 = 30\geq25).
  • For (5x + 12y): (5\times50+12\times20=250 + 240=490<500). So yes, she could have sold 50 small and 20 large pictures.