Create a word problem with two linear equations. One equation is a proportional relationship and the other is not a proportional relationship. Interpret the meaning of the slope and y-intercept in context.
A farmer is planning to start a new garden. He knows he needs to buy both seeds and fertilizer to maintain his garden. The price of seeds is $2 per packet, while the price of fertilizer is $10 per bag. The farmer wants to buy a total of 20 items for his garden and is willing to spend a maximum of $60.
Let's represent the cost of seeds as 2s and the cost of fertilizer as 10f, where s is the number of seed packets and f is the number of bags of fertilizer.
The total cost of seeds and fertilizer should not exceed $60, which can be represented by the equation:
$2s + $10f ≤ $60
This inequality represents a relationship that is not proportional because the cost does not increase at a constant rate. The slope of this equation would be 2/10 = 0.2, which represents the rate at which the cost of seeds increases relative to the cost of fertilizer.
Now, let's consider the number of seed packets the farmer can buy based on his budget. Since he wants to buy a total of 20 items, we can also represent this relationship as:
s + f = 20
This equation represents a proportional relationship because the total number of items (seeds and fertilizer) depends on a fixed ratio of 1:1. The slope of this equation would be -1, indicating that for every additional bag of fertilizer purchased, the farmer would need to buy one less packet of seeds to maintain a total of 20 items.
In this context, the slope of the proportional relationship represents the fixed ratio of seeds to fertilizer needed to maintain a balanced garden, while the y-intercept represents the initial cost of purchasing seeds and fertilizer before exceeding the total budget of $60.