Let H be a subgroup of G. If G has exactly one subgroup of order |H|, then show that for all g e G, gH = Hg. Hint: First, show that g Hg- is a subgroup of G where ge G then use this fact.

Dominique makes and sells pies. The number of pies demanded weekly, qD, is given by qD = 75 – 3p where p is the price of a pie (in $).  The number of pies she is willing to supply weekly, qS, is given by qS = 4p – 20 where p is the price of a pie (in $).  If the price of a pie is set at $15, how much of a shortage or surplus of pies would there be in a week?  Blank 1. Fill in the blank, read surrounding text.   Write either shortage or surplus in the box. of  Blank 2. Fill in the blank, read surrounding text. pies

1. Which situation is best represented by the equation? X – 4 = 16? Alisa picked 16 apples and ate of them. What is x, the number of apples she had left? Jose ran for 16 minutes and walked for 4 minutes. What is company x, the difference between the time he spent running and the time he spent walking? James spent $4 of his allowance and had $16 left. What is x, the total amount of Jordan's allowance? Annette has hit 4 of the last 16 balls pitched. What is x, the total number of balls pitched?

3. Solve the system of equations. -4x + 3y = -18 X – 4y = 11

1. Solve 2x = -20 2. Find 2 solutions to 2x > -20

A bank teller has a total of 73 bills in five-, ten-, and twenty-dollar denominations. The total value of the money is $810. (a) Find the total number of solutions. (b) Find the solution with the smallest number of five-dollar bills. (c) Find the solution with the largest number of five-dollar bills. (a) Let x be the number of five-dollar bills, y be the number of ten-dollar bills, and z be the number of twenty-dollar bills. Translate the given information into two equations, using x, y, and z as the variables. x+y+z = 73 5x + 10y + 20z = 810 (Do not factor. Use integers or decimals for any numbers in the expressions.) There are a total of solutions.