**Can you think about 2 different examples where you can formulate 2 different linear equations?**

**Can you think about 2 different examples where you can formulate 2 different linear equations?**

## Response 1

Wow! So this is a pretty intense question. My answer would go something like this: A customer goes into a cell phone store and buys a case for his new I-phone for $20.00. However, the store just sold the last $20 case so they will need to ship the case to the customer. Shipping is free but they since the customer is shipping to mexico it would cost an additional $2. My equation would look like this: x=20y+2, y=2/20 and y=2. The second scenario would look like this: Karl likes cherries. he bought a container of cherries for 8. He tipped the farmer 2 since the farmer did not use any pesticides. The equation would look like this: y=8x+2 y=2/8 x=4## Response 2

My answer would be through incidents that happened in my life:**1)**I have two nieces and a nephew, all born in September. I tend to buy their gifts early, since they visit my mom’s house in the summer and it’s easier that way. This year (since they are getting older), I was planning on buying them Beats headphones (so that they can keep up with the teenage Joneses). I went to the store and saw that the Beats are on sale for $119 each. I also have to ship them to the house individually so that they all can feel special and open their own gift. That is a fixed cost of $45 total. So, my equation would look like: y=119x+45. x= -45/119 and y=45.

**2)**When I was younger my uncle ran a barber shop and charged $11,x, per haircut for adults. He would give my sister and I $2 to sweep the floor for him everyday. We thought we were so cool. In any case, my uncle’s barber shop equation should look something like: y=11x-2, where y=-2 and x =2/11.

## Response 3

EXAMPLE 1
Recently, I needed plumbing done in my home, and I was told that the plumber would charge a $50 flat fee. For every additional hour that he stayed, he would charge an additional $30. I realized that I can write this as an equation where Y is the total amount that I owe to the plumber, and X represents the number of labor hours that he puts in. So we have:
Y = 30X + 50
To find the Y-intercept, I set X = 0, and then solve for Y.
Y = 30(0) + 50
Y = 0 + 50
Y = 50
So then my Y-intercept is at Y = 50.
To find the X-intercept, I set Y = 0, and then solve for X.
0 = 30X + 50
-50 = 30X
X = -50/30
X = -5/3
So then my X-intercept is at X = -5/3
The positive slope (the slope is 30) implies that I will owe the plumber more and more money the more hours that he works. If the slope were, negative, that wouldn’t make sense. It would be more like he’s giving me money, but I won’t complain!!
EXAMPLE 2
For my son’s birthday party, I decided to give out goldfish to all the the children. I started out with 100 goldfish, and each hour, I would give out approximately 12 goldfish. In this case, I can let Y be the number of goldfish that I have remaining, and X will be the number of hours that have passed. So we have:
Y = -12X + 100
My slope is -12 because I am actually losing fish after each hour. If for some reason I was gaining fish, then my slope would be positive.
To find the Y-intercept, I set X = 0, then solve for Y.
Y = -12(0) + 100
Y = 0 + 100
Y = 100
So then my Y-intercept is at Y = 100.
To find the X-intercept, I set Y = 0, and then solve for X.
0 = -12X + 100
-100 = -12X
X = 100/12
X = 25/3
So then my X-intercept is at X = 25/3
It’s a good thing the slope is negative, because I don’t want any of those fish back!

The steepness of the line is represented numerically by the slope of the line. It is the quotient of the change in y and the change in x.

The x intercept is 4 x, y 4,0 this is the point where the graph intersects the x-axis. The y-coordinate of this point is 0 because it is on the x-axis.
3

The x-intercept is 5. The y-intercept is -2.5.
x-2(0) = 5
x = 5
0-2y = 5
-2y=5 -y=5/2 y= -2.5
The slope is .5

## Response 4

**1)**In a grocery store, you can use linear equations to forecast and find break-even amounts. In forecasting profits, a grocery store may know that when their products are on sale, the amount of products sold is linearly related. When a box of cereal is sold for $6 at the regular price, the store averages 60 boxes sold in a week. When the store reduces the price to $4 a box, the average number of boxes sold in the week increases to 90. The linear equation is as follows: (6, 60); (4, 90) M = (60 – 90)/(6-4) = (-30)/2 = -15 (y – 90) = -15(x-4) Y = -15x +60 + 90 Y = 150 – 15x**2)**The grocery store can also use linear equations for a break-even analysis. The store has a fixed cost of $200 a week for the cereal and each box of cereal costs $2 to make. The regular price of the cereal is $6 a box. To figure out how many boxes must be sold each week to break even, the following equation can be solved: 6x = 200 + 2x 6x – 2x = 200 4x = 200 X = 50 boxes of cereal must be sold per week to break even## Response 5

I want to pull from my current work environment (an assisted living community) for my examples. We have several departments with different situations, procedures, and goals, but as many other students have proven in this discussion, these equations exist in in more ways than we initially think. I am not sure that these are accurate applications in word problem form, but here goes nothing! Example 1: My community has eight new hires attending orientation. There are two managers, both of whom will need one polo shirt for special occasions only. There are six front-line employees, each of whom will need three polo shirts to wear as part of their uniforms. x = number of polo shirts y = number of employees receiving specified quantity of shirts (1, 2) (3, 6) m = (6-2) / (3-1) = 4 / 2 = 2 (y-2) = 2(x-1) y-2 = 2x – 2**y = 2x**x intercept = 0 y intercept = 0 Example 2: Our 2013 rate was $1,500/month with 200 apartments occupied. Our 2014 rate is $1,550/month with 150 apartments occupied. x = Rent rate y = Number of apartments occupied ($1500, 200) ($1550, 150) m = (150-200) / (1550-1500) = (-50) / (50) = -1 (y-200) = -1(x-1500) y = -1x + 1700 x intercept = 1700 y intercept = 1700## Response 6

For my first example: I am in a band of with three other people, and we do cover songs. We recently received an invitation to play at very popular bar on a crowded Friday night. The owner of the bar has promised that as the main band, that we will earn $20 for each hour that we play. While the hours that we decide to play may change, the amount of money offered does not. Therefore, we could look at the equation as: y=20x For my second example: A coworker and I have a bet going on. As part of our job, we have to make calls to customers in order to send out information. While trying to make some fun out of a fairly monotonous task, my manager decides to give us five dollars for every successfull call we have to make. Putting that into an equation, we could look at the relationship like this: 5x+5b= y## Response 7

Two different examples of 2 different linear equations, including the x-intercept and y-intercept of each linear equation. Slope and interpretation of slope is given.**1)**With a $12 budget, Sam goes to Lowe’s to buy 3 plants for the garden and 4 packages of seed. The plants available vary in price from $1 to $4. The seed packages vary in price from free $0 [a garden club promotional giveaway] to $2.25. The combinations of plants and seed packages are shown below, to allow Sam to stay within his budget yet get still plants and/or seed for his garden. 3x + 4y = 12x | y | 3x + 4Y |

4 | 0 | 12 |

3 | 0.75 | 12 |

2 | 1.5 | 12 |

1 | 2.25 | 12 |

slope |

0.75 |

*x*+ 4( 0 ) = 12 put 0 in the y value in the equation 3*x*+ 0 = 12 3*x*= 12*x*= 12/3*x*= 4 The y intercept is 3 x,y 0,3 this is the point where the graph intersects the y-axis. The x-coordinate of this point is 0 because it is on the y-axis. 3( 0 ) + 4*y*= 12 0 + 4*y*= 12 4*y*= 12*y*= 12/4*y*= 3**2)**Larry wants to go out to his favorite restaurant to get something to eat after work. The menu has appetizers available ranging from $5 to $9. He has a coupon that the restaurant emailed to him that will give him $1 off if he buys $7 of food, $2 off if he buys $9 of food. The coupon does not include the $5 appetizer. It is double the value of the coupon day. Larry only has $5 available, plus some extra for a tip for the waiter. His options for ordering are shown below. x- 2y = 5x | y | x-2y |

7 | 1 | 5 |

9 | 2 | 5 |

0 | -2.5 | 5 |

5 | 0 | 5 |

## Response 8

Example 1 On Sunday I to the mall to buy shoes for $48.00, because it was father’s day weekend there was a 20% off discount off the regular price. What is the regular price of the shoes? Let x be the regular price. A 20% discount is 0.2 x, so the discounted price is x – 0.2x. Set this equal to 48 and solve for x: $48= x – 0.2 $48 = 100/100-20% $48 = 0.8x 48/0.8=0.8x/0.8 X= $60 X-intercept = 60 Y-intercept = 0 (60 ,0) Example 2 On my Sunday shopping spree I spent $705 for two pairs of shoes and three bags. If the shoes costs $40 more than the bag, How much was the shoes and bags? Let us assume the cost of the bag to be x. 2(40 + x) + 3x = 705 80 + 2x + 3x = 705 80 + 5x = 705 5x = 705 – 80 5x = 625/5 x = 125 40 + x = 40 + 125 = 165 The cost of each bag is $125 and that of each pair of shoes is $165. X-intercept = 125 Y-intercept =165 (125,165)## Response 9

First Example: The cost of washing a car (y) consists of $1.50 in soap (fixed cost) and $.20 per gallon for water used (variable cost x). Thus the equation would be: y = $.20x + $1.50 where x = the number of gallons of water used Since y = mx + b where b = the y-intercept Thus the y-intercept = 1.50 To find the x-intercept, set y = 0 0 = $.20x + $1.50 Thus the x-intercept is: x = -7.5 Since the slope = m, m = 1/5 Second Example: To build one’s own pizza with just one topping, the cost (y) is broken down as follows: The combined fixed cost of the dough, sauce, one topping and cheese equals to $6. Any additional toppings cost $.50 a piece (variable cost x). Thus the equation would be: y = $.5x + $6 where x = the number of toppings requested Since y = mx + b where b = the y-intercept Thus the y-intercept = 6 To find the x-intercept, set y = 0 0 = $.5x + $6 Thus the x-intercept is: x = -12 Since the slope = m, m = 1/2## Response 10

**1)**My best friend is getting married in August and planning his wedding reception. He plans to have an open bar, but will need to hire 2 bartenders for every 50 guests that will be at the reception. X will represent the number of guests in increments of 50 and Y will be the total number of bartenders he will need to hire. 2 bartenders times every 50 guests(x) will equal total bartenders to hire(y). 2X=Y X Intercept 2x=0; 0/2=x; X=0 Y Intercept 2(0)=y; Y=0 X Y 2 4 3 6 4 8 6 12 (8-6)/(4-3)= 2 (slope)**2)**He and his wife-to-be decided they will likely have around 200 guests (x=4 from previous equation) so they will need to hire 8 bartenders. They will have to pay each bartender $10/hour for each hour they decide to have the open bar, which will cost them a total of $80/hour ($10 for each bartender x 8 per hour = $80). The number of hours the bar will be open(x) times $80/hr will equal their total cost for the bartenders. 80X=Y X Intercept 80x=0; 0/80=x; X=0 Y Intercept 80(0)=y; Y=0 X Y 3 240 4 320 5 400 6 480 (320-240)/(4-3) = 80 (slope) Both equations listed are positive slopes because as x increases, so does y in both scenarios. The second scenario offers a steeper slope than the first scenario, which means it has a great absolute value than the first scenario, and also a greater rate of change in the dependent variable as the independent variable changes.