NAME:
DATE:
Unit 1 – Using Linear Systems to Solve Problems
Modelling with Linear Equations
Many real-life relationships between data, such as costs to start and run a business
can be modelled using linear equations. These mathematical models let us make
predictions about relationships. Sometimes a model only involves one equation,
but for many relationships the model involves two or more equations.
This type of model is called a system of linear equations.
independent
variable
the variable whose value determines that of the dependent
variable
dependent
variable
the variable whose value depends on the value of the
independent variable
e.g. y = x + 2
is the independent variable
is the dependent variable
Air Kool Ltd. repairs air conditioners and charges a flat fee of $15.00,
plus a labour fee of $35.00/h. Hank Verdi charges a flat fee of $25.00
plus $25.00/h.
1. Create a table of values for both Air Kool and Hank for repairs ranging from 0 h to 5 h.
Air Kool
Time (h) Cost ($)
0
1
2
3
4
5
Hank
Time (h) Cost ($)
0
1
2
3
4
5
2. Describe any patterns you see in the tables.
3. What two quantities are related in each repair situation?
4. In each situation, identify the dependent variable and the independent variable.
Represent these two quantities using two different variables.
5. Use the variables to write an equation that models the repair charges for:
a)
Air Kool
b)
Hank
6. Refer to your table of values. Identify the ordered pair that satisfies both equations.
7. What do these coordinates mean in terms of this problem?
Translating English to Mathematics
Certain words indicate certain mathematical operations.
Practice:
Place the following words under the column of the correct operation.
together
combined
more than
less than
Addition
decreased by
increased by
product of
quotient
minus
total of
yields
times
Subtraction
percent
will be
out of
gives
Multiplication
less
are
sum
was
is
of
Division
Equals
Examples:
Write each sentence as a mathematical equation with two unknowns (variables).
Remember to include a “let” statement.
1. The sum of two numbers is 12.
2. It costs $135 to rent the car, based on $25 per day, plus $0.15 / km.
3. The total number of adults and children at the circus was 1254.
4. Al invested some money at 8% and some at 10%. He earned a total of $235 in interest.
Additional Exercises:
1. Two numbers are represented by x and y. Translate the following into algebra.
a) the sum of the two numbers is 15
b) double the first number plus triple the second number has a total of 25
c) the sum of one-half the first number and one-third the second number gives 19
2. Greg has x $2 coins and y $5 bills. Translate the following into algebra.
a) the total value of the bills yields $35
b) the total number of coins and bills is 10
3. Jess has n nickels and d dimes. Translate the following into algebra.
a) the value of the nickels, is 35 cents
b) the value of the dimes, is 90 cents
c) the total value of the coins gives 190 cents
d) the total number of coins is 19