Ryan S.

asked • 12/09/20

What is the slope intercept form equation and how many donations do you need?

You need to create a valentines package. 50 valentine packages cost $4300. You've started your own charity donation program. The three scenarios listed below show how much donations(y) you've collected in(x) days.

Given three ordered pairs:

Scenario 1: (6, 305); 6 days, you've collected $305.00

Scenario 2: (14, 585); 14 days, you've collected $585.00

Scenario 3: (0, 125); at the start of the donation program, you had $125.00 saved up



Items are price times 50.


So,

Flowers: $20 * 50 = $1000

Candy: $20 * 50 = $1000

Card: $12 * 50 = $600

Gift: $18 * 50 = $900

Dinner: $16 * 50 = $800

Total Expense for 50 valentines packages = $4300


Based on the three scenarios, write the equation in slope intercept form.



Using the amount of money you need to create 50 valentine packages, how many donations do you need?


Karina F.

Hi. Can you check the values given in the 3 scenarios? For one linear equation to be applicable for all the ordered points given, the slope must be the same for the line based on the ordered points. If you compute the slope, m for each two ordered points, the slopes are not the same. Example, slope between (6, 305) and (0, 125) is 30/1. The slope between (14, 585) and (6, 305) is 35/1. So one general equation will not apply to all 3 scenarios.
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12/09/20

Ryan S.

Thank you! I will let my instructor know.
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12/09/20

Ryan S.

Also, when you write the equation in slope intercept form, the question asks how many donations are needed. However, isn’t the equation you write representing how much you earned in donations? This makes sense to me because I see it as you tell me how many days I’ve collected donations and I can tell you how much in dollars worth of donations I’ve collected, which makes sense because y is dependent on x(the number of days). So how are we able to find the amount of donations needed?
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12/09/20

Karina F.

You are correct in your interpretation of the what the final equation represents. The question (to me) is to find out how many days (of receiving donations) are needed to break even and earn back the upfront cost of the packages and initial $125 spent. The slope is essentially the rate of money donated per day, and you need this to solve for x. The slope between (6, 305) and (0,125) is 30/1. That means that every day $30 are donated. Now that $30/day can be looked at as one donation/day with a value of $30. Almost like an hourly rate at a job. So in 6 days...donations went up ($30/day) for a total of $180...but you can look at it as 6 donations ($30/donation) in 6 days each donation averaging $30. So x can be looked as just days but you can also look at it as the # of donations to make the $180. Does that make sense? That is why the slope has to make sense and be constant in order to make ONE equation that applies to all 3 scenarios. That is my interpretation.
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12/09/20

1 Expert Answer

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Stu M. answered • 12/10/20

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Part 1


The slope (m) is the rate $/days. This can be found by using any two of the ordered pairs of points given in the problem. Let's choose the first two.


(6, 305)

(14, 585)


If the relationship is linear, we can find the slope by using m = change in y / change in x:

m = (585 - 305) / (14 - 6)

m = 280 / 8

m = 35 $/day


The y-intercept (b) can be found by determining the starting point of the graph, or where the line intercepts the y-axis. This is identified in the problem:


Scenario 3: (0, 125); at the start of the donation program, you had $125.00 saved up


Therefore, b = $125 because there is $125 saved up at zero days.


We can plug this into slope-intercept form to find the equation of the line:

y = mx + b

y = 35x + 125

y = (35 $/day) * x + $125


x and y represent days and donations here, so we can see that for every day (x), the charity earns $35 and the charity starts at $125.


Part 2


We need $4300 to create 50 valentines. In order to find the number of days to raise this amount, we can plug y = $4300 into the equation we just found.

y = 35x + 125

4300 = 35x + 125

4175 = 35x

x = 119.3


You can't determine the number of donations needed, because we don't know how much each donation is in dollars. However, we have found that we need 119.3 days of fundraising to get to $4300. In other words, after 120 days, we will have enough for the 50 valentine packages.

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Ryan S.

When finding the slope, why would you not include (0, 125)? I took the starting point of when you began collecting donations where you saved $125. Then used 14 days after you’ve collected donations (14, 585) as the end point. I also arrived at the slope of $35 per day but that is only including scenario 1 and 2 and not 3. But yet we are using the starting point from scenario 3. In other words, why are we using only scenario 1 and 2 to find the slope yet we are using scenario 3 as our y intercept? Shouldn’t they all be included to find each piece? Thanks for your help!
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12/10/20

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