SOLVING MULTI-STEP EQUATIONS��(SPECIAL SOLUTION SETS)�
Math 8
King Midas thinks his children are brilliant. They have had so much success in determining how many coins are in the bags of the gifts he leaves for them in the treasury. We will return one final time to his kingdom to assist the royal children in their task, but King Midas has a surprise in store for them. Every week he is giving the Prince and Princess the same number of coins. Draw a picture for each scenario and use it to create an equation to solve for the number of coins in a bag.
Week 1: Princess Perfect has two bags and 12 gold coins. Prince Charming has three boxes, inside a box is one bag and one gold coin. How many coins are in each bag?
Princess
Prince
=
Open the boxes
Then set aside the same number of
bags and coins from each child.
Week 1: Princess Perfect has two bags and 12 gold coins. Prince Charming has three boxes, inside a box is one bag and one gold coin. How many coins are in each bag?
Princess
Prince
Set aside the same number of
bags and coins from each child.
=
=
Each bag contains 9 coins
Open the boxes
Week 2: Prince Charming has seven bags and 15 coins. Princess Perfect has five bags and 18 coins. How many coins are in each bag?
Princess
Prince
Set aside the same number of
bags and coins from each child.
=
Week 2: Prince Charming has seven bags and 15 coins. Princess Perfect has five bags and 18 coins. How many coins are in each bag?
Princess
Prince
Set aside the same number of
bags and coins from each child.
=
Each bag has One and a half coins
=
Week 3: Today there are so many coins in the treasury and thus some boxes were needed to hold the riches. The Prince has two boxes, each box holds four bags and six gold coins. Princess Perfect has eight bags and 12 gold coins. How many coins are in each bag?
Princess
Prince
Open the boxes
Then set aside the same amount of
bags and coins from each child.
Week 3: Today there are so many coins in the treasury and thus some boxes were needed to hold the riches. The Prince has two boxes, each box holds four bags and six gold coins. Princess Perfect has eight bags and 12 gold coins. How many coins are in each bag?
Princess
Prince
=
There are no bags
remaining for either child
How many coins
could be in a bag?
The possibilities are endless…
Week 4: The kingdom is prosperous! Prince Charming finds he has three boxes and in each box there are three bags and two gold coins. The Princess has no boxes, only nine bags and 27 gold coins. How many coins are in each bag?
Prince
Princess
Then set aside the same amount of bags
and coins from each child.
Open the boxes
Week 4: The kingdom is prosperous! Prince Charming finds he has three boxes and in each box there are three bags and two gold coins. The Princess has no boxes, only nine bags and 27 gold coins. How many coins are in each bag?
Prince
Princess
The Prince has
nothing remaining
=
≠ 0
No matter how many coins in a bag they won’t be equal
Steps for Solving Multi-Step Equations:
1. Use the Distributive Property to eliminate parentheses and combine any like terms on the same side of the equation
-there should be at most two terms on either side
3. Use inverse operations to solve the 2-step problem if variables remain
4. Check your solution!!
2. Use addition or subtraction to move all variables to one side of the problem
- sometimes all the variables are eliminated, this creates a special solution set
Solve each equation
Check your solution!
Replacing the variable with the found solution
resulted in both sides being equal
Solve each equation
Where is the variable?
What is the value of y ?
Special Solution Set #1:
If the variables disappear and
you are left with a true statement,
All Real Numbers
will be the solution.
Exact same on
both sides
True Statement
Solve each equation
Check your solution!
Where is the variable?
False Statement
What is the value of p ?
Solve each equation
Special Solution Set #2:
If the variables disappear and you
are left with an untrue statement,
there is No Solution
to the equation
Solve each equation
All Real Numbers
Solve each equation
No Solution
How do you know it is a special solution set?
How can you tell which special set of solutions it is?
If the coefficient on the variables is exactly the same
(look at the number on the left of the variable)
IF the constants are same
ALL REAL NUMBERS
IF the constants are different
NO SOLUTION