1. Students will learn the concept of the substitution method of solving systems of equations.
2. Students will accurately solve systems using substitution.
3. Students will apply the substitution method to symbolize and solve real-world problems.
Digital projector connected to teacher's laptop (or photocopy of attached example)
Attached example word problem
At the beginning of the lesson, ask students to get with their partners and solve the following system of equations (placed on the board):
x + 2y = 4
and 3x – y = 5
Circulate through the room, redirecting and reminding where necessary. This will serve as review not only of the initial definition of a system of equations, but also of the mechanics of rapid graphing and slope. After 3-5 minutes, bring the students back together as a class to discuss results. (The correct answer is (2, 1).) Address any misconceptions or difficulties.
Next, project the attached online example. Ask the student pairs to represent the problem by defining variables and finding equation(s) that the problem represents. Remind them that they are not trying to SOLVE the problem, only to set it up (i.e. define all variables and find equations that represent the situation).
Use some of the following questions as guiding questions:
How many different kinds of variable quantities are there? (Elicit that there are 2 – granola bars and drinks)
What should we use to represent these quantities? (Elicit: two different variables.)
What kind of relationships do we have between the quantities of these variables? (Elicit that their total equals 120.)
What kind of relationship is there between the price of the items and the total amount of money made (revenue)? (Elicit that price times quantity equals total amount made on each item, and that their sum should give the total revenue.)
Use the discussion to arrive at the following system of equations:
x + y = 120
and 2x + 4y = 280
Now, lead a brief discussion to demonstrate that accurate graphing would be fairly unreasonable to solve this system. Remind students of the essential skill in Algebra – packing skills! What does that mean? Substitute a variable in the place of a complex expression. Notice, if I modify the first equation to the following format:
x = 120 – y
I have created an expression that I can pack into another variable. In other words, for every x
, I can "pack in" 120 –y
. On the board, show that you can substitute one expression in for the variable:
2 (120 –y ) + 4y = 280
Have students solve this equation for y
and then use the value for y
to find the value for x
. Have them check their answers in the original equations and against the problem itself. (The solution is (100, 20) -- 100 granola bars and 20 drinks.)
Provide the following example on the board and ask students to try to use this same method with their partners:
y = x – 3
4x + y = 32 Solution: (7, 4)