Question

The flip flop a lot company makes and sells flip flops they have one linear function that represents the cost of producing flip-flops and another linear function that models how much income they get from those flip-flops describe the key features that will determine if these linear functions ever intercepted

Answer 1

If the linear functions have the same slope but different y-intercepts, they will never intercept. If the If the linear functions have the same slope and the same y-intercet, they are the same line. If the linear functions have different slopes and different y-intercepts, they will intercept. 

Answer 2

Calling  

y1: cost of producing flip-flops  

y2: income for selling flip-flops  

x: number of flip-flops produced/sold  

The cost of producing flip-flops is modeled as:  

y1 = m1*x + b

where the slope (m1) and y-intercept (b) are positive, so that, the cost is always positive (notice that x>0)

The income for selling flip-flops is modeled as:  

y2 = m2*x

For the same reason as before, m2 is positive. Here there is no b, because if  no units are sold (x = 0) then no income is made (y = 0).

In order to incomes overcome costs, m2 must be greater than m1. For x = 0 y1 is greater than y2 (remember the absence of b in y2 equation), but given that m2>m1, then y2 grows faster than y1 and eventually they intercept. At this point costs are equal to incomes, before this point costs are greater than incomes, and after this point incomes are greater than costs. So,  this point represent the minimum number of flip-flops the company must sell if it don't want to go bankrupt.