Module 4
Solving Equations I
Lesson 3
Start
solving equations i
Alice made straight A's this year and her hard work in school really paid off. As a reward for her effort, her grandpa gave her a $100 gift card. Now Alice can purchase the jacket she wanted for $20 and a new video game for $35. But how will she spend the rest?
To figure out the remaining balance, Alice needs to write and solve an algebraic equation. She needs to be sure she’s right so she doesn’t overspend the gift card.
Goals
Objectives: - define and apply inverse operations of addition or subtraction.
- solve one-variable equations containing addition or subtraction.
- solve real-world problems by writing and solving equations.
Solving Equations I-scenario
Alice received the $100 gift card from her grandpa for getting straight A's in school. She wants to purchase a jacket for $20 dolllars and a video game for $35. She wants to make sure she figures out how much money she'll have left on her card after her purchases.
cost $35.00
cost $20.00
gift card $100.00
The cost for the video game and jacket need to be part of the equation. The total amount Alice has to spend is $100.00. Will she have money left over on her gift card? How do you know? What is needed to complete the equation?
Questions:
Will there be money left on the gift card after her purchases?
How do you know? What is needed to complete the equation?
exploration
Equations
Before you can help Alice solve equations , you must know how to write them correctly. You already know how to translate expressions, and translating real-world problems into equations uses a similar skill set.
Revisit!
Thinking about the Solution
There are multiple strategies for solving equations. Explore below for other ways to solve x + 20 = 55.
Bar Model
Fact Families
Fact families review
drag and drop the numbers to complete fact families for each set
Balancing is important
So far you have learned how to use reasoning, fact families, and bar models to solve an equation for the solution. Having a variety of strategies will help you understand how to help Alice determine the remaining balance on her gift card.
Another strategy you can use is to think of equations as a balance scale. Let’s see how it works.
3+3=6
Balancing scale practice
x+4=7
x+3=5
click
click
practice:
Complete these problems on your own before checking your answer.
Use the fact families to determine the solution to the equation:
x − 9 = 22
click for answer
Use the balance scale to determine the solution to the equation x + 3 = 7. A bag plus 3 blocks has the same weight as 7 blocks. What can you do to both sides of the scale to determine the weight of the bag only?.
x+3=7
click for answer
inverse operations
A balancing scale can be helpful when you need to visualize how to solve an equation, but it can be limiting too. Sometimes numbers will be too big, or equations will have lots of operations.
You can also use inverse operations to solve an equation. Inverse means opposite. When you have an operation and an inverse operation together, they will “undo” each other.
For example, 10 + 3 − 3 still equals 10. Taking away 3 units “undoes” adding 3 units, because the result is like adding a zero.
inverse operations
Example 3:
Example 2:
Example 1:
Solve x - 13 = 58 using inverse operations.
Show that 8 + 4 - 4 is equal to 8.
Solve y + 4.93 = 15.06 using inverse operations.
Show Answer
Show Answer
Show Answer
Solving Equations-scenario solution
Recall: Alice received a $100 gift card with which she wants to purchase a jacket for $20 dolllars and a video game for $35. She wants to figure out how much money will be left on her card after her purchases.
cost $35.00
cost $20.00
gift card $100.00
total left on gift card
35.00 + 20.00 + x = 100.00 55.00 + x = 100.00 -55.00 -55.00 x = $45.00
Combine like terms on either side before isolating x.
The inverse of +55.00 is -55.00.
We must also -55.00 from the right side to keep it balanced.
Remember: "Tally up" after each step, then determine what to do next.
There will be $45.00 left on Alice's gift card after she purchaces the video game and jacket.
check your understanding
Question 1/3
What is the first step in solving the equation x - 16 = 42?
Determining the inverse of -16 and performing that operation on both sides of the equation.
Subtracting 16 from 42 to determine the value of x.
correct answer
The goal is always to isolate the variable, which we can do by determining the correct inverse operation needed.
check your understanding
Question 2/3
Solve for y: y + 3 = 12.25
y=15.25
y = 9.25
correct answer
y = 9.25
check your understanding
Question 3/3
Solve for m:m - 3.45 = 15.60
m = 15.60
m = 19.05
m = 12.15
correct answer
oh, oh!
This answer is not correct...
Try again!
Good job
Lesson completed
Home
Our goal is to isolate x
x - 13 = 58 + 13 +13 x + 0 = 71 or simply... x = 71
+13 is the inverse of -13
We must also +13 to the right side of the equation to keep it balanced!
It is important to stay organized. After each step of an equation, "tally up" to determine what you have left before moving forward. In this case, we have the answer, but in more complex equations, you may need another step.
y + 4.93 = 15.06 - 4.93 -4.93 y + 0 = 10.13 or simply... y = 10.13
Our goal is to isolate y
-4.93 is the inverse of +4.93
We must also -4.93 to the right side of the equation to keep it balanced!
Remember: "Tally up" to determine what you have left after each step. In this case, we have the answer, but in more complex equations, you may need another step.
8 - 4 + 4 8 + 0 8
Subtracting 4 and adding 4 are inverse operations
When the inverse operations of addition and subtraction are used, the resulting value is zero.
When solving equations involving addition or subtraction, it is important to remember that they are inverses.
Addition to "undo" subtractionSubtraction to "undo" addition
35 + 15 = 60 ----FALSE45 + 15 = 60 ----TRUE55 + 15 = 60 ----FALSE
You can isolate the bag by removing 4 blocks from each side, which leaves the bag on the left side and 3 blocks on the right side. The scale is balanced, so the bag must weigh the same as 3 blocks. The solution to the equation x + 4 = 7 is x = 3.
Solving Equation I-Module 4 Lesson
Middle School
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Transcript
Module 4
Solving Equations I
Lesson 3
Start
solving equations i
Alice made straight A's this year and her hard work in school really paid off. As a reward for her effort, her grandpa gave her a $100 gift card. Now Alice can purchase the jacket she wanted for $20 and a new video game for $35. But how will she spend the rest? To figure out the remaining balance, Alice needs to write and solve an algebraic equation. She needs to be sure she’s right so she doesn’t overspend the gift card.
Goals
Objectives:- define and apply inverse operations of addition or subtraction.
- solve one-variable equations containing addition or subtraction.
- solve real-world problems by writing and solving equations.
Solving Equations I-scenario
Alice received the $100 gift card from her grandpa for getting straight A's in school. She wants to purchase a jacket for $20 dolllars and a video game for $35. She wants to make sure she figures out how much money she'll have left on her card after her purchases.
cost $35.00
cost $20.00
gift card $100.00
The cost for the video game and jacket need to be part of the equation. The total amount Alice has to spend is $100.00. Will she have money left over on her gift card? How do you know? What is needed to complete the equation?
Questions:
Will there be money left on the gift card after her purchases? How do you know? What is needed to complete the equation?
exploration
Equations
Before you can help Alice solve equations , you must know how to write them correctly. You already know how to translate expressions, and translating real-world problems into equations uses a similar skill set.
Revisit!
Thinking about the Solution
There are multiple strategies for solving equations. Explore below for other ways to solve x + 20 = 55.
Bar Model
Fact Families
Fact families review
drag and drop the numbers to complete fact families for each set
Balancing is important
So far you have learned how to use reasoning, fact families, and bar models to solve an equation for the solution. Having a variety of strategies will help you understand how to help Alice determine the remaining balance on her gift card. Another strategy you can use is to think of equations as a balance scale. Let’s see how it works.
3+3=6
Balancing scale practice
x+4=7
x+3=5
click
click
practice:
Complete these problems on your own before checking your answer.
Use the fact families to determine the solution to the equation:
x − 9 = 22
click for answer
Use the balance scale to determine the solution to the equation x + 3 = 7. A bag plus 3 blocks has the same weight as 7 blocks. What can you do to both sides of the scale to determine the weight of the bag only?.
x+3=7
click for answer
inverse operations
A balancing scale can be helpful when you need to visualize how to solve an equation, but it can be limiting too. Sometimes numbers will be too big, or equations will have lots of operations. You can also use inverse operations to solve an equation. Inverse means opposite. When you have an operation and an inverse operation together, they will “undo” each other. For example, 10 + 3 − 3 still equals 10. Taking away 3 units “undoes” adding 3 units, because the result is like adding a zero.
inverse operations
Example 3:
Example 2:
Example 1:
Solve x - 13 = 58 using inverse operations.
Show that 8 + 4 - 4 is equal to 8.
Solve y + 4.93 = 15.06 using inverse operations.
Show Answer
Show Answer
Show Answer
Solving Equations-scenario solution
Recall: Alice received a $100 gift card with which she wants to purchase a jacket for $20 dolllars and a video game for $35. She wants to figure out how much money will be left on her card after her purchases.
cost $35.00
cost $20.00
gift card $100.00
total left on gift card
35.00 + 20.00 + x = 100.00 55.00 + x = 100.00 -55.00 -55.00 x = $45.00
Combine like terms on either side before isolating x.
The inverse of +55.00 is -55.00.
We must also -55.00 from the right side to keep it balanced.
Remember: "Tally up" after each step, then determine what to do next.
There will be $45.00 left on Alice's gift card after she purchaces the video game and jacket.
check your understanding
Question 1/3
What is the first step in solving the equation x - 16 = 42?
Determining the inverse of -16 and performing that operation on both sides of the equation.
Subtracting 16 from 42 to determine the value of x.
correct answer
The goal is always to isolate the variable, which we can do by determining the correct inverse operation needed.
check your understanding
Question 2/3
Solve for y: y + 3 = 12.25
y=15.25
y = 9.25
correct answer
y = 9.25
check your understanding
Question 3/3
Solve for m:m - 3.45 = 15.60
m = 15.60
m = 19.05
m = 12.15
correct answer
oh, oh!
This answer is not correct...
Try again!
Good job
Lesson completed
Home
Our goal is to isolate x
x - 13 = 58 + 13 +13 x + 0 = 71 or simply... x = 71
+13 is the inverse of -13
We must also +13 to the right side of the equation to keep it balanced!
It is important to stay organized. After each step of an equation, "tally up" to determine what you have left before moving forward. In this case, we have the answer, but in more complex equations, you may need another step.
y + 4.93 = 15.06 - 4.93 -4.93 y + 0 = 10.13 or simply... y = 10.13
Our goal is to isolate y
-4.93 is the inverse of +4.93
We must also -4.93 to the right side of the equation to keep it balanced!
Remember: "Tally up" to determine what you have left after each step. In this case, we have the answer, but in more complex equations, you may need another step.
8 - 4 + 4 8 + 0 8
Subtracting 4 and adding 4 are inverse operations
When the inverse operations of addition and subtraction are used, the resulting value is zero.
When solving equations involving addition or subtraction, it is important to remember that they are inverses.
Addition to "undo" subtractionSubtraction to "undo" addition
35 + 15 = 60 ----FALSE45 + 15 = 60 ----TRUE55 + 15 = 60 ----FALSE
You can isolate the bag by removing 4 blocks from each side, which leaves the bag on the left side and 3 blocks on the right side. The scale is balanced, so the bag must weigh the same as 3 blocks. The solution to the equation x + 4 = 7 is x = 3.