Algebra
Solving Equations I
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.
The cost for the video game and jacket are needed 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 eqution?
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. Just locate the words that indicate the equal sign.
Revisit!
Thinking about the Solution
drag and drop the numbers to complete fact families for each set
You just solved an algebraic equation using one strategy; however, there are other strategies you can use. Let’s take a look at the other tools you can use to help solve the equation x + 20 = 55.
Bar Model
Fact Families
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.
Practice
Please complete to check your understanding
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?.
click for answer
inverse operation
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 operation
Example 3:
Example 1:
Example 2:
Use the mathematical properties to prove that 8b + 0 − b is equivalent to 8b.
Use the mathematical properties to prove that 2(x + 5 + 7x) is equivalent to 16x + 10.
Show that the expression 4x + 1 + 3x + 8 is equivalent to 7x + 9. Can you identify which properties are used to show that the two expressions are equivalent? Think about which property allows you to move from step to step.
click to enlarge
Show how they are equivalent
Show how they are equivalent
Show with substitution
Show with substitution
Show with substitution
Show another proof
Show with another number
Be very careful to not rely on the substitution method as your only method to prove two equations are equivalent. This method should be used in addition to your use of other mathematical properties. This is because it is possible that two expressions can be equal for one number, but not another.
Combining like terms
Combining like terms means combining the coefficients of the like terms. The purpose of combining like terms is to help simplify an expression.
Take a look at the examples below to understand what it means to combine like terms:
combine like terms
check your understanding
Question 1/3
Are x² and x like terms?
True
False
correct answer
x² and x are not like terms.
check your understanding
Question 2/3
Are 3x² and 6x² like terms?
True
False
correct answer
3x² and 6x² are like terms.
check your understanding
Question
Which expression is equivalent to 2(3x + 4 + x)?
click on an expression
x + 8+ 2x
8x + 8
7x + 4
correct answer
2(3x + 4 + x) Use the distributive property (6x + 8 +2x) Combine like terms 8x + 8
vocabulary
inverse operation
An operation that reverses the effect of another operation; for example, adding three and subtracting three are inverse operations.
example
equation
A mathematical sentence that shows two expressions are equal using the equal sign.
example
summary
Proof Expressions are Equivalent
Like and Unlike Terms
Proof using mathematical properties and simplifying:8t + 6p + 10t + 8p = 8t + 10t + 6p + 8p Use the commutative property. = (8 + 10)t + (6 + 8)p Use the distributive property. = 18t + 14p Combine like terms.= 18t + 14p Proof using substitution: 18t + 14p 8t + 6p + 10t + 8p = 18(1) + 14(1) 8(1) + 6(1) + 10(1) + 8(1) = 18 + 14 8 + 6 + 10 + 8 = 32 32
Like terms have identical variables but may have different coefficients. When terms have vari ables that are different they are called unlike terms.
Combining Like Terms
Combining like terms means to combine the coefficients of the like terms. Remember, the coefficient of a variable includes the number and the sign in front of the number. The purpose of combining like terms is to simplify an expression. Simplifying makes it easier to evaluate the expression and to prove if it is equivalent to another expression.
Proving Expressions Are Equivalent
In an algebraic expression, you can use the mathematical properties to group and combine like terms. The properties also allow you to prove that the simplified expression is equivalent to the beginning expression. Example: Prove the expression 8t + 6p + 10t + 8p is equivalent to the expression 18t + 14p.
Good job
Lesson completed
Reflect on what you have learned in this module.
Home
oh, oh!
This answer is not correct...
Try again!
click for audio
click for audio
35 + 15 = 60 ----FALSE45 + 15 = 60 ----TRUE55 + 15 = 60 ----FALSE
click for audio
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.
click for audio
Solving Equations I
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Transcript
Algebra
Solving Equations I
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.
The cost for the video game and jacket are needed 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 eqution?
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. Just locate the words that indicate the equal sign.
Revisit!
Thinking about the Solution
drag and drop the numbers to complete fact families for each set
You just solved an algebraic equation using one strategy; however, there are other strategies you can use. Let’s take a look at the other tools you can use to help solve the equation x + 20 = 55.
Bar Model
Fact Families
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.
Practice
Please complete to check your understanding
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?.
click for answer
inverse operation
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 operation
Example 3:
Example 1:
Example 2:
Use the mathematical properties to prove that 8b + 0 − b is equivalent to 8b.
Use the mathematical properties to prove that 2(x + 5 + 7x) is equivalent to 16x + 10.
Show that the expression 4x + 1 + 3x + 8 is equivalent to 7x + 9. Can you identify which properties are used to show that the two expressions are equivalent? Think about which property allows you to move from step to step.
click to enlarge
Show how they are equivalent
Show how they are equivalent
Show with substitution
Show with substitution
Show with substitution
Show another proof
Show with another number
Be very careful to not rely on the substitution method as your only method to prove two equations are equivalent. This method should be used in addition to your use of other mathematical properties. This is because it is possible that two expressions can be equal for one number, but not another.
Combining like terms
Combining like terms means combining the coefficients of the like terms. The purpose of combining like terms is to help simplify an expression. Take a look at the examples below to understand what it means to combine like terms:
combine like terms
check your understanding
Question 1/3
Are x² and x like terms?
True
False
correct answer
x² and x are not like terms.
check your understanding
Question 2/3
Are 3x² and 6x² like terms?
True
False
correct answer
3x² and 6x² are like terms.
check your understanding
Question
Which expression is equivalent to 2(3x + 4 + x)?
click on an expression
x + 8+ 2x
8x + 8
7x + 4
correct answer
2(3x + 4 + x) Use the distributive property (6x + 8 +2x) Combine like terms 8x + 8
vocabulary
inverse operation
An operation that reverses the effect of another operation; for example, adding three and subtracting three are inverse operations.
example
equation
A mathematical sentence that shows two expressions are equal using the equal sign.
example
summary
Proof Expressions are Equivalent
Like and Unlike Terms
Proof using mathematical properties and simplifying:8t + 6p + 10t + 8p = 8t + 10t + 6p + 8p Use the commutative property. = (8 + 10)t + (6 + 8)p Use the distributive property. = 18t + 14p Combine like terms.= 18t + 14p Proof using substitution: 18t + 14p 8t + 6p + 10t + 8p = 18(1) + 14(1) 8(1) + 6(1) + 10(1) + 8(1) = 18 + 14 8 + 6 + 10 + 8 = 32 32
Like terms have identical variables but may have different coefficients. When terms have vari ables that are different they are called unlike terms.
Combining Like Terms
Combining like terms means to combine the coefficients of the like terms. Remember, the coefficient of a variable includes the number and the sign in front of the number. The purpose of combining like terms is to simplify an expression. Simplifying makes it easier to evaluate the expression and to prove if it is equivalent to another expression.
Proving Expressions Are Equivalent
In an algebraic expression, you can use the mathematical properties to group and combine like terms. The properties also allow you to prove that the simplified expression is equivalent to the beginning expression. Example: Prove the expression 8t + 6p + 10t + 8p is equivalent to the expression 18t + 14p.
Good job
Lesson completed
Reflect on what you have learned in this module.
Home
oh, oh!
This answer is not correct...
Try again!
click for audio
click for audio
35 + 15 = 60 ----FALSE45 + 15 = 60 ----TRUE55 + 15 = 60 ----FALSE
click for audio
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.
click for audio