Introduction to Modelling
Linear systems are useful when you are trying to solve for the value
of an unknown in situations that can be represented by lines.
Ex1 . Bert earns an hourly wage plus tips. One week he worked 12h and made a total of 117$. The next week he worked 10h and earned the same amount in tips as the week before, for a total of 110$. What is Bert's hourly wage?
∎ identity the dependent and independent
variables, and the constants.
let
let
let
let
t represent the number hours Bert worked (indep.)
E represent how much Bert earns (dep.)
r represent Bert's hourly wage (constant)
T represent tips received (constant between
the two weeks)
∎ state the relationship between these values.
earnings = (hourly wage) (# of hours worked) + (Tips)
E = (r)(t) + T
∎ Input the specific information found in the question.
"One week he worked 12h and made a total of 117$"
117 = (r)(12) + T
117 = 12r + T
"The next week he worked 10h and earned the same amount in tips as the week before, for a total of 110$."
110 = (r)(10) + T
110 = 10r +T
∎ Solve for the required value.
In this case, solve for r.
Ex 2. The sum of two numbers is 72. Their difference is 48. Find the numbers.
Ex 3:
Alex drove 500 km from Windsor to Peterborough in 5 1/2 h. He drove part of the way at 100 km/h, and the rest of the way at 80 km/h. How far did he drive at each speed?
∎ distance = (speed) x (time)
d
s
t
Ex 4. A boat took 2 h to travel 24 km down a river with the current and 3 h to make the return trip against the current. Find the speed of the boat in still water and the speed of the current.
then solve for s and c
HW: p. 93 # 14
p. 103 # 17
A. Kareem took 5 hours to drive 470km from Sudbury to Brantford. For part of the trip, he drove at 100 km/h. For the rest of the trip, he drove at 90km/h. How far did he drive at each speed?
*B. It took a patrol boat 5 hours to travel 60km up a river against the current, and 3h for the return trip with the current. Find the speed of the boat in still water and the speed of the current.