Linear Equation

Question

Adele mows lawns in her neighborhood each weekend to save money for college. She uses her parent’s lawn mower, but is responsible for buying gas and other supplies she'll need. The table below shows her profit after the first two lawns she mowed last weekend.

Adele charges the same amount for every lawn she mows. Create a function that models the relationship between her profit, 𝑃, and the number of lawns she mows, 𝑚.

Answer 1

For a linear equation, y = mx + b where this m is the slope

From the table, we can generate two equations

m + b = 10 (Eq 1)

2m + b = 35 (Eq 2)

Subtracting Eq 2 from Eq 1 to eliminate b.

2m + b = 35

-m - b = -10

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m = 25 Which is the slope for the equation (not the number of lawns mowed, confusing, huh?)

and substituting m = 25 into Eq1:

25 + b = 10 ---> b = -15 (The expenses for the weekend)

So the model would be

P(x) = 25x - 15

or

P(m) = 25m - 15 to used m as the number of lawns instead of x, the same form as the table.

Answer 2

You would use the table to set up two "coordinates" (x, y), or (m,P) in this case, like this:

1. (1,10), (2, 35) Then calculate the slope m = (y - y₁)/(x - x₁)
2. (35-10)/(2-1) = m = 25
3. y = 25x + b now use one of the two points to put in for the x and y and solve for b
4. 10 = 25(1) + b
5. b = -15
6. y = 25x - 15 (now switch the x and y for m and P as the problem requested).
7. P = 25m - 15