We are looking for the linear equation P(n) = mn + b where n is equal to the number of shirts sold and P(n) is the price per shirt.
We know that if n = 3,000, P(n) = 78, and if n = 7,000. P(n) = 62. We can write these in quation form (using P(n) = mn +b):
[tex]\begin{gathered} 78=3,000m+b \\ 62=7,000m+b \end{gathered}[/tex]We can then use the substitution method to solve for m and b. Let's use 78 = 3,000m + b to express b.
[tex]\begin{gathered} 78=3,000m+b \\ 78-3,000m=b \end{gathered}[/tex]We'll then substitute this in 62 = 7,000m + b.
[tex]\begin{gathered} 62=7,000m+b \\ 62=7,000m+(78-3,000m) \\ 62-78=7,000m-3,000m \\ -16=4,000m \\ m=-\frac{16}{4,000}=-0.004 \end{gathered}[/tex]Then we can use m = -0.004 to solve for b.
[tex]\begin{gathered} b=78-3,000b \\ b=78-3,000(-0.004) \\ b=75+12 \\ b=87 \end{gathered}[/tex]We now have the complete equation for P(n):
[tex]P(n)=-0.004n+87[/tex]