We are asked to write a system of equations which could be used to find the total cost c of renting a canoe for n hours. We will create two equations: one will represent the cost of renting a canoe on River A, and the other on River B. Let's start with River A.
RiverACost:?Let's say that to rent a canoe on River A, we need to pay $6 fee and for each hour of rental we need to pay an extra $3. This means that the total cost c of renting a canoe for n hours is $6 plus $3 multiplied by n.
RiverACost: c= 6+ 3* n
The canoe on River B could have a higher initial fee — for example, $18. However, the hourly rate could be a bit lower. One hour of rental could cost $1. Thus, on River B the total cost c of renting a canoe for n hours would be $18 plus $1 multiplied by n.
RiverBCost: c= 18+ 1* n
We created a system of two equations, and each can represent the cost of renting a canoe for n hours.
c=6+3* n c=18+1* n
This is only an example of a system! Yours might be different.
We created a system of equations that represents the costs of renting a canoe on two rivers.
c=6+3* n c=18+1* n
Both equations are in slope-intercept form, so we can easily identify the slope and the initial value of each equation.
River
Equation
Initial Value
Slope
River A
c= 6+ 3* n
6
3
River B
c= 18+ 1* n
18
1
Let's sketch a graph of River A's equation. The initial value of the equation is 6. Also, the slope is equal to 3, which means that as the x-values increase by 1 the y-values increase by 3.
Now, let's sketch the graph of River B's equation. This time, the initial value is 18. The slope is equal to 1, which means that the x- and y-values increase by the same amount.
We graphed both equations!
In Part B, we graphed the equations that represents the prices of renting a canoe on two rivers.
We can see that that the lines intersect at exactly one point.
The lines intersect at coordinates n= 6 and c= 24. This means that after 6 hours the total cost for renting a canoe will be the same on both rivers, $24. Note that if the lines on your graph do not intersect, this might mean that the cost will never be the same.
