15% as a decimal is 0.15
Multiply the amount of the bill by the percentage:
54.83 x 0.15 = 8.22
The tip was $8.22
The equation of the line is x=4
Answer: average means add all scores and divide by number of scores earned
last score is x
so the sum is 75+81+x
3 scores so (75+81+x)/3=average
must be no less than (greater than or equal to 80)
(75+81+x)/3≥80
solve
times both sides by 3 to clear fraction
75+81+x≥240
156+x≥240
minus 156 from both sides
x≥84
the studen must earn at least 84 on last exam
Step-by-step explanation:
![\bf x^2y^2+3xy=1\\\\ -------------------------------\\\\ \stackrel{\stackrel{\textit{test for x-symmetry}}{y=-y}}{x^2(-y)^2+3x(-y)=1}\implies x^2y^2-3xy=1\impliedby \begin{array}{llll} \textit{function differs}\\ \textit{from original}\\ \textit{no dice} \end{array}\\\\ -------------------------------\\\\](https://tex.z-dn.net/?f=%5Cbf%20x%5E2y%5E2%2B3xy%3D1%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%5Cstackrel%7B%5Cstackrel%7B%5Ctextit%7Btest%20for%20x-symmetry%7D%7D%7By%3D-y%7D%7D%7Bx%5E2%28-y%29%5E2%2B3x%28-y%29%3D1%7D%5Cimplies%20x%5E2y%5E2-3xy%3D1%5Cimpliedby%20%0A%5Cbegin%7Barray%7D%7Bllll%7D%0A%5Ctextit%7Bfunction%20differs%7D%5C%5C%0A%5Ctextit%7Bfrom%20original%7D%5C%5C%0A%5Ctextit%7Bno%20dice%7D%0A%5Cend%7Barray%7D%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C)
![\bf \stackrel{\stackrel{\textit{test for y-symmetry}}{x=-x}}{(-x)^2y^2+3(-x)y=1}\implies x^2y^2-3xy=1\impliedby \begin{array}{llll} \textit{function differs}\\ \textit{from original}\\ \textit{no dice} \end{array}\\\\ -------------------------------\\\\ \stackrel{\stackrel{\textit{test for origin-symmetry}}{x=-x~~y=-y}}{(-x)^2(-y)^2+3(-x)(-y)=1}\implies x^2y^2+3xy=1\impliedby \begin{array}{llll} origin\\symmetry \end{array}](https://tex.z-dn.net/?f=%5Cbf%20%5Cstackrel%7B%5Cstackrel%7B%5Ctextit%7Btest%20for%20y-symmetry%7D%7D%7Bx%3D-x%7D%7D%7B%28-x%29%5E2y%5E2%2B3%28-x%29y%3D1%7D%5Cimplies%20x%5E2y%5E2-3xy%3D1%5Cimpliedby%20%0A%5Cbegin%7Barray%7D%7Bllll%7D%0A%5Ctextit%7Bfunction%20differs%7D%5C%5C%0A%5Ctextit%7Bfrom%20original%7D%5C%5C%0A%5Ctextit%7Bno%20dice%7D%0A%5Cend%7Barray%7D%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%5Cstackrel%7B%5Cstackrel%7B%5Ctextit%7Btest%20for%20origin-symmetry%7D%7D%7Bx%3D-x~~y%3D-y%7D%7D%7B%28-x%29%5E2%28-y%29%5E2%2B3%28-x%29%28-y%29%3D1%7D%5Cimplies%20x%5E2y%5E2%2B3xy%3D1%5Cimpliedby%20%0A%5Cbegin%7Barray%7D%7Bllll%7D%0Aorigin%5C%5Csymmetry%0A%5Cend%7Barray%7D)
so, recall, the function has symmetry when the yielded resulting function resembles the original function, after negativizing the variable(s).
Also recall that minus*plus is minus, and minus*minus is plus.
See https://web2.0calc.com/questions/help_55278.