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A clothing business finds there is a linear relationship between the number of shirts, n , it can sell and the price, p , it can charge per shirt. In particular, historical data shows that 19000 shirts can be sold at a price of $ 34 , while 24000 shirts can be sold at a price of $ 14 . Give a linear equation in the form p = m n + b that gives the price p they can charge for n shirts

Answer :

Linear equation is p = (-0.004) n + 110

Step-by-step explanation:

We have, p = price, n = no. of shirts , m = slope,

b = constant (y - intercept)

linear equation as , [tex]\boldsymbol{p}=(\boldsymbol{m} \times \boldsymbol{n})+\boldsymbol{b}[/tex]                         .....(1)

(p,n)= ($34, 19000) & ($14, 24000)

putting above values in equation (1) we get,

⇒ 34 = (19000×m) + b ⇒  [tex]m=\frac{(34-b)}{19000}[/tex]            .... (2)

and, 14 = (24000×m) + b ⇒  [tex]m=\frac{(14-b)}{24000}[/tex]          ....(3)

Equating equations (2) & (3),

⇒ [tex]\frac{(14-b)}{24000}=\frac{(34-b)}{19000}[/tex]

⇒ [tex]\frac{(14-b)}{24}=\frac{(34-b)}{19}[/tex]

⇒ 19 (14 – b) = 24 (34 – b)

⇒ [tex](19 \times 14)-19 b=(24 \times 34)-24 b[/tex]

⇒ [tex]24 b-19 b=-(19 \times 14)+(24 \times 34)[/tex]

⇒ [tex]5 b=-(19 \times 14)+(24 \times 34)[/tex]

⇒ [tex]5 \mathrm{b}=-266+816=550[/tex]

⇒ [tex]\mathrm{b}=\frac{550}{5}=110[/tex]

⇒ b = 110

Now, [tex]m=\frac{(14-110)}{24000}=\frac{-96}{24000}=-0.004[/tex]

putting values of m & b in equation (1) we get :

      [tex]p=(-0.004) n+110[/tex]

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