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3.7 Solving by Substitution

Last updated 10 months ago
32 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
  • Solve systems of equations using the substitution method
  • Explore special cases of systems of equations
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP4
  • CCSS.HSF.IF.B.4
  • CCSS.HSF.IF.B.5
Personal Finance Objectives
  • Explore simple interest savings accounts
National Standards for Personal Financial Education
Saving
  • 2a: Select a preferred location for a savings account based on comparison of interest rates and fees at different types of financial institutions.
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
OBJECTIVES & STANDARDS
Math Objectives
  • Solve systems of equations using the substitution method
  • Explore special cases of systems of equations
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP4
  • CCSS.HSF.IF.B.4
  • CCSS.HSF.IF.B.5
Personal Finance Objectives
  • Explore simple interest savings accounts
National Standards for Personal Financial Education
Saving
  • 2a: Select a preferred location for a savings account based on comparison of interest rates and fees at different types of financial institutions.
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
Intro- Warm-Up
GRAPH: The Problem with Graphing
You have to make a decision about how to spend your grocery budget. Here are your options:
  • You can continue shopping at your local grocery store where you will spend $100 per week on essential groceries
  • You can get a membership at your local Costco where you will pay a $55 membership fee but will pay only $80 per week for groceries
2
Part I
Write an equation for each situation where x represents the number of weeks and y represents total cost.
Grocery: _______
Costco:_______
2

Part II
  1. Graph your equations on the given grid below

  • Click the graph tab.
  • Click on the graph background to add a point. Add two points to create a graph. Drag a point or type in x and y coordinates to edit its position. Click on a point to delete it.
1
Part III
During what week do these situations have the same value?_______ week(s)
1

Is your answer exact? If yes, explain how you found an exact answer. If no, explain why you had difficulty finding an exact answer.

Learn It
Solving Systems by Substitution
While graphing can be used to show the intersection of two lines, it has its limitations. If we are looking for a precise answer, we can calculate the solution using algebra.
The substitution method allows you to calculate the solution of a system by solving one equation for a variable, then substituting that equation into the other equation of your system.
  1. Review the example problem completed below.




1
Isolate one variable: Solve one equation for either variable (if necessary)



2
Substitute:
Substitute that solved equation into the other equation.



3
Solve: There should only be one variable remaining. Solve that for that variable.





4
Plug in:
To find the other variable, plug your solution into one of the original equations and solve.





5
Check it:
Check your solution by plugging into BOTH equations of the system to be sure that you get a true statement








6
Answer:
Write your answer as a point (x, y)
(4, -1)
1
For each problem, solve the system of equations using the substitution method.


_______
1


_______
1

_______
1

_______
1

When solving by substitution, how do you choose which variable to solve for?

* Paper Copy of Student activity Sheet, Pages 4 & 5:


1
y = 2x + 1 2y + x = 7
_______
1
y = -x + 4 y = 3x
_______
1
y = 3x - 10 2y = x - 5
_______
1
2y = x + 1
-2x - y = 7
_______
1
3y = 2x 2y + x = 7
_______
1
3x - 2y = 2 5x - 5y = 10
_______
1
5x + 4y = -14 3x + 6y = 6
_______
Learn it (page 5)
1

Special Cases
Samantha has $250 in her bank account and makes $15 per hour.
Whitney has $500 in her bank account and makes $15 per hour.
  1. When will Samantha and Whitney have the same amount of money?

4
If we look at a graph of this situation, we can verify that the lines never intersect:
Samantha: y = 15x + 250
Whitney: y = 15x + 500

What does this look like algebraically?
Samantha: _______
Whitney: _______
Use the substitution method to solve this system of equations._______
What conclusion can you draw about how systems of equations with no solution will work out algebraically?_______
8
Practice Solving Systems of Equations From Word Problems
Let’s solve some more systems but now you’ll have to write equations that represent the situation first!  Here are some helpful tips to get you started:
  • Start with the question.  What are you being asked to find?  Those are your variables!  Remember that systems will have two unknowns.
  • A system of equations will have two equations so you will have to find two situations in your word problem that relate to your unknown variables.
Example 1
You are deciding between First National Bank and Liberty Bank.  They both offer checking and savings accounts.  First National charges a $10 fee to receive checks  and charges $2.50 per ATM transaction.  Liberty Bank has no check fee but charges $3.25 per ATM transaction.  How many times do you need to be charged an ATM fee before you are paying the same amount of money in fees at both banks?
How to Solve:
Step 1: Define your variables -Look at the question statement. What are you being asked to find?
x=_______
y=_______
Step 2: Write 2 equations that use those unknown variables:
First National Bank Fees: _______
Liberty Bank Fees: _______
Step 3: Solve the system of Equations:
x=_______
Step 4: State your solution. After_______ ATM fees, you will be paying the same amount of money in fees.
Step 5: Reflect: Does it make sence to have a decimal answer for this situation? _______ How would you change your answer. _______
2
Marcus has 15 coins in his piggy bank consisting of only nickels and quarters. The total value of the coins is $1.95. How many of each coin does he have in the bank?
Nickels: _______
Quarters: _______
2
When you get insurance on your home, you have the option to add extra insurance for very valuable items like jewelry and art. You find that your insurance company charges a flat fee plus a percentage of the value of the item. It costs $39 to insure a $1200 and $77 to insure a $3100 item. What are the flat rate _______ and percentage_______% charges?
Application Problems
APPLICATION: Solving by Substitution for Savings Goals

Level 1

Building an Emergency Fund
You are working two part-time jobs. Working at the local library pays $15 per hour and working as a personal trainer pays $20 per hour. Last week you made $800 and worked 45 total hours.
4
Find out how many hours you worked at each job.
a. Write a system of equations to represent the scenario. Remember to define your variables as:
x=hours worked at the library
y=hours worked as a personal trainer
Equation for Total Hours Worked: _______
Equation for Total Income: _______

b. Solve the system of equations. How many hours did you work at each job? Library (x)=_______
Personal Trainer (y)=_______
2
You decide to use the 50/30/20 rule and devote all of your savings towards building an emergency fund of $10,000. Assume you earn the same amount each week.
  1. How much do you save towards your emergency fund each week?_______
  2. How many weeks will it take to save $10,000?_______
1
How many fewer weeks would it take to save for your emergency fund if you could work all 45 hours as a personal trainer? It would take_______ fewer weeks to save $10,000
Simple Interest
Remember the simple interest equation I=Prt where P represents principal balance, r represents the interest rate as a decimal, and t represents time. If you need a refresher on using the simple interest formula, click here!

Malik deposits $350 into a simple interest savings account that earns 2% per year. Saraiah deposits $400 into a different simple interest savings account that earns 1% per year.
1
How much does Malik earn in interest in one year? Use the simple interest formula._______ dollars
1
How much does Saraiah earn in simple interest per year? Use the simple interest formula._______ dollars
2
Write an equation for each account where x represents the number of years that have passed and y represents total interest earned plus principal.
Malik: _______
Saraiah: _______
1
How long will it take for Malik and Saraiah to have the same amount of money?_______ Years
Turning a Profit
A break-even analysis is used to figure out how many units of a product need to be sold in order to pay expenses (aka breaking even) and turn a profit. This can be done by writing two equations:
  1. The cost function which represents fixed costs (usually one time costs) and variables costs (costs that can change, usually based on the number of items produced)
  2. The revenue function which represents money earned from selling the product
You open a business to make and sell custom bracelets. You purchase office equipment and a laptop to market and sell your product which costs $1150. The cost to manufacture one bracelet is $2.40 and you plan to sell them for $11.
0
Define your Variables:
x=Number of bracelets
y=Total Cost/Revenue
0
Write a system of equations to represent cost and revenue.
Cost: _______ Revenue: _______
0
Solve your system to calculate how many bracelets you will need to sell to break even and start turning a profit._______bracelets
0
Profit is the money you make when you subtract costs from revenue. Write an equation that represents the profit that you would make from selling a certain number of items._______