The SneakerRama Company makes and sells sneakers. They have one linear function that represents the cost of producing sneakers and another linear function that models how much income they get from those sneakers. Describe the key features that would determine if these linear functions ever intercepted.

- anonymous

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- anonymous

@Michele_Laino @mathstudent55 @mathmate

- Michele_Laino

any linear function can be described by this formula:
\[y = kx + h\]
where k and h are coefficients, and x, and y are two variables

- Michele_Laino

now, in your first case y is the cost of producing sneakers
and for second linear function y is the income for sneakers

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## More answers

- Michele_Laino

what can be x?

- anonymous

um..the sneakers?

- Michele_Laino

yes! the number of sneakers

- anonymous

Yay:) Now what?

- Michele_Laino

we have to consider one functionat a time.
So for first function if x=0, we have:
y=h, namely if we have not produced sneakers then the cost is h

- anonymous

Ok, i see

- anonymous

so what would be the key features to determine if they intersected?

- Michele_Laino

I rewrite the two functions as below:
\[\begin{gathered}
f\left( x \right) = {a_1}x + {b_1} \hfill \\
g\left( x \right) = {a_2}x + {b_2} \hfill \\
\end{gathered} \]

- Michele_Laino

we have intersection if:
\[\begin{gathered}
f\left( x \right) = g\left( x \right) \hfill \\
{a_1}x + {b_1} = {a_2}x + {b_2} \hfill \\
\left( {{a_1} - {a_2}} \right)x = {b_2} - {b_1} \hfill \\
\end{gathered} \]

- anonymous

alright, but how would I type that in word form?

- Michele_Laino

and we have no solutions, if and only if:
\[\large {a_1} - {a_2} = 0,\;{b_2} - {b_1} \ne 0\]

- Michele_Laino

namely, if the producing rate is equal to the earning rate and the cost for producing no sneakers is different from the earning for no sneakers sold, then we have no intersection

- anonymous

ok but we have to determine the key features if they ever intersceted?

- anonymous

"Describe the key features that would determine if these linear functions ever intercepted." See?

- Michele_Laino

yes!

- anonymous

so what would be the answer?

- anonymous

Can I ask another question please?

- Michele_Laino

I think that the requested key can be this:
the producing rate has to be different from the earning rate

- anonymous

OK:)

- Michele_Laino

:)

- anonymous

Your boss hands you the monthly data that show the number of orders coming in to and out of the warehouse. The data are in the table below. Explain to your boss, in complete sentences, the solution to this system and what the solution represents.
Month No. of orders in No. of orders out
January (1) 3 3
February (2) 6 4
March (3) 9 5
April (4) 12 6

- anonymous

# of orders in:
3
6
9
12

- anonymous

#of orders out:
3
4
5
6

- Michele_Laino

we can write the relationship between the orders out y, as a function of the orders in x, like below:
\[\Large y = \left( {x - 1} \right) + 3\]

- anonymous

ok:)

- Michele_Laino

please wait

- anonymous

Alright:P)

- Michele_Laino

no, I my formula is wrong

- Michele_Laino

my fromula is wrong

- anonymous

that's okay:)

- anonymous

we can always start over

- Michele_Laino

we have two linear equations. Namely the relationship between months x and orders out f(x), which is:
\[\Large f\left( x \right) = \left( {x - 1} \right) + 3\]

- Michele_Laino

the second relationship is between the months x and the orders in g(x):
\[\Large g\left( x \right) = 3x\]

- anonymous

ok:) I see

- Michele_Laino

the solution of your system is given solving this equation:
\[\Large \begin{gathered}
f\left( x \right) = g\left( x \right) \hfill \\
\\
\left( {x - 1} \right) + 3 = 3x \hfill \\
\end{gathered} \]
solve please, for x

- anonymous

ux x- I got x+2=3x, then I got 2x+2

- Michele_Laino

I got:
2x=2

- Michele_Laino

so, x=?

- anonymous

1

- Michele_Laino

that's right!

- Michele_Laino

it means that at first month the orders in are equal to the orders out

- Michele_Laino

more precisely:
the number of orders in is equal to the number of orders out

- anonymous

Thanks! S :P)

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