1. Problems leading to systems of equations

2. The substitution method
Marco thinks that there is only one way to solve the following system of equations:
3x – y = 9
8x + 5y = 1
Is Marco correct? Solve the system of equations using substitution.
Now solve the equations using the elimination method instead.
Which method is better / easier?
Teacher notes
Using the substitution method:
Rearrange the first equation to y = 3x – 9. Now substitute y = 3x – 9 into the second equation: 8x + 5(3x – 9) = 1
This simplifies to 23x = 46, so x = 2.
Substitute x = 2 into the first equation, to find y: 3 × 2 – y = 9, so y = –3.
Ensure that students check the solution by substituting back into the first equation.
Using the elimination method:
Multiply the first equation by 5, so it becomes 15x – 5y = 45.
Add the two equations together to eliminate y: 15x + 8x – 5y + 5y =
We now have the equation: 23x = 46. As in the substitution method, we then find x = 2 and y = –3. Students should check the solution by substituting back into an original equation.
There is no method that is "better" than the other. However, students might find one method easier than the other and should be encouraged to use whichever method they find the easiest.
Mathematical Practices
3) Construct viable arguments and critique the reasoning of others.
Students should state whether or not they agree with Marco and explain why/why not. If not, they should be able to demonstrate both methods.
Systems of equations can be used to solve problems in every day life.

3. Solving problems
Archie is trying to figure out how old his two math teachers are!
Mr. Addison and Miss Peters have given Archie two clues.
The sum of their ages is 76.
Mr. Addison is 22 years older than Miss Peters.
Teacher notes
While it would be possible to solve this problem using trial and error, students should be able to see that creating a system of equations is a far neater and simpler way of solving this problem. Using trial and error with these clues would be an educated estimate and it wouldn’t take too long to work out the correct answer, but the algebraic method is a more efficient method.
Call the Mr. Addison’s age a and Miss Peters’ age p. Use the information to write a two equations in terms of a and p:
a + p = 76
a – p = 22
Adding these equations gives: 2a = 98 so a = 49
Substituting a = 49 into the first equation gives: 49 + p = 76 so p = 27
We can check these solutions by substituting them into the second equation: a – p = 22 giving 49 – 27 = 22
This is true and so the solution is correct. Mr. Addison is 49. Miss Peters is 27.
Mathematical Practices
3) Construct viable arguments and critique the reasoning of others.
Students should state whether or not they agree with Archie and explain why/why not. If not, they should be able to demonstrate the best method.
4) Model with mathematics.
This slide shows how systems of equations can be written and solved to help with real-life situations.
How old is each teacher?
Archie thinks the best method of solving the problem is by trial and error. Do you agree? Justify your answer.

4. Price juggling
The Gonzalez family and the Harris family are taking a trip to the circus.
In the Gonzalez family, there are 4 adults and 3 children.
The total cost of their tickets is $95.
In the Harris family, there are 2 adults and 6 children.
The total cost of their tickets is $88.
Mathematical Practices
4) Model with mathematics.
This activity provides a real-life example of how systems of equations can be used to find unknown quantities. Students should be able to write two equations describing the cost of each family’s tickets, then solve them to answer the questions in context.
Photo credit: © InnervisionArt, Shutterstock.com 2012
Write a system of equations to describe the costs.
How much does an adult’s ticket cost? How much does a child’s ticket cost?

5. Price juggling Gonzalez family: 4 adults and 3 children, costing $95.
Harris family: 2 adults and 6 children, costing $88.
Let a = adult’s ticket cost and c = child’s ticket cost.
4a + 3c = 95 2a + 6c = 88 A
×2: 4a + 12c = 176 B C C
– : A
9c = 81
c = $9
Mathematical Practices
1) Make sense of problems and persevere in solving them.
Students should be able to see that by writing two equations both involving the unknown values, they will have a system of equations that can be solved. Their answers can be verified by substituting back into one of the original equations.
2) Reason abstractly and quantitatively.
Students should be able to take these real-life situations and write equations to describe them, choosing appropriate letters to represent unknown quantities. They should be able to manipulate these equations (using the elimination or substitution method) to solve for the unknowns, remembering to answer with the correct units, e.g. "an adult ticket costs $8.50".
4) Model with mathematics.
This activity provides a real-life example of how systems of equations can be used to find unknown quantities. Students should be able to write two equations containing two unknown quantities and solve them to answer the questions in context.
Photo credit: © mitya73, Shutterstock.com 2012
Substitute c = 9 into equation to find a: B
2a + 6(9) = 88
subtract 54:
2a = 34
Check the prices by substituting back into equation
divide by 2:
a = $17 A