8.1 - Addition and Subtraction of Algebraic Expressions
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Introduction to Algebraic Expressions
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Welcome, everyone! Today, we’re going to explore the exciting world of algebraic expressions. Can anyone tell me what an algebraic expression is?
Is it a combination of numbers, variables, and operations?
Exactly! For instance, in the expression `2x + 3`, `2x` is a term that combines a coefficient with a variable. Now, let's discuss how we can add them together. What do we mean by 'like terms'?
Like terms are terms that have the same variable parts!
Great job! When we add like terms, we combine their coefficients. For example, if we have `3x + 2x`, we can simply add `3 + 2` to get `5x`. Let’s recap: if you understand like terms, adding expressions becomes easy!
Adding Algebraic Expressions
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Now, let’s move to adding algebraic expressions. If I ask you to add `4x + 3` and `5x - 2`, what would be your first step?
I think we should line up like terms!
"Exactly! Aligning like terms helps with clarity. Let’s write it down:
Subtracting Algebraic Expressions
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Next, let’s explore subtraction. Can someone explain why subtracting a term is akin to adding its opposite?
Because subtracting a positive is the same as adding a negative?
"Exactly! For instance, to subtract `5x` from `2x`, we change it to addition:
Combining Adding and Subtracting
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Now, let’s practice combining both addition and subtraction. If I give you `3x + 5 - 2x + 4`, how do we solve it?
We first rewrite it as `3x - 2x + (5 + 4)`.
Absolutely! And what does that simplify to?
It simplifies to `x + 9`.
Well done! Remember, you can always combine constants before focusing on the variables. This method will help simplify more complex expressions!
Practice Problems and Recap
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Alright class, let’s solidify your understanding with some practice. What is the sum of `2a + 3b - c` and `4a - 2b + 5c`?
That would be `6a + b + 4c`!
And if we subtract `5a - 2b + c` from that, we get `a + 3b + 3c`.
Correct! Always remember to align your terms properly, and apply those signs carefully. Let’s summarize: addition combines like terms while subtraction involves flipping signs. Keep practicing!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section provides a detailed exploration of adding and subtracting algebraic expressions through examples, emphasizing the alignment of like terms. It demonstrates practical methods for performing operations on various expressions, ultimately clarifying the rationale behind algebraic manipulations.
Detailed
Addition and Subtraction of Algebraic Expressions
In this section, we delve into the essentials of adding and subtracting algebraic expressions, which serve as the foundation for algebraic manipulation and solving equations. An algebraic expression consists of variables, constants, and operators. Common examples include expressions like x + 3, 2y - 5, and 3x^2.
Key Concepts
- Like Terms: Terms that contain the same variables raised to the same power. For instance, in the expression
4xy + 5xy, both terms are like terms because they both contain the variablesxandy. - Adding Expressions: To add multiple expressions, align like terms vertically to simplify the addition process. For example:
- Adding
7x^2 - 4x + 5with9x - 10results in:
7x^2 - 4x + 5
+ 9x - 10
---------------
7x^2 + 5x - 5
- Subtracting Expressions: Subtraction is equivalent to adding the additive inverse. Students will learn that subtracting a term means adding its opposite. For example, subtracting
5becomes adding-5.
Significance
Understanding these operations allows students to manipulate algebraic expressions effectively, setting the stage for more complex algebraic concepts, including solving equations and functions.
Youtube Videos
Audio Book
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Understanding Algebraic Expressions
Chapter 1 of 5
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Chapter Content
In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x^2, 4xy + 7 etc.
Detailed Explanation
Algebraic expressions are combinations of numbers, variables, and operations (like addition or subtraction). They can represent mathematical relationships and can be as simple as a single term like x or as complex as 4xy + 7. The key is that they involve at least one variable.
Examples & Analogies
Imagine you have a box that can hold a varying number of items. You can express the number of items in the box as x, which can change. If you add some fixed items, like 3, the total can be expressed as x + 3, showing how the contents of the box can vary.
Adding Algebraic Expressions
Chapter 2 of 5
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Chapter Content
To add 7x^2 – 4x + 5 and 9x – 10, we do:
7x^2 – 4x + 5 + 9x – 10
This results in 7x^2 + 5x – 5. Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown.
Detailed Explanation
When adding algebraic expressions, it’s important to line up like terms vertically. Like terms are terms that have the same variable raised to the same power. For example, – 4x and 9x are like terms, and you can add their coefficients: – 4 + 9 = 5. You also add constant terms together, such as 5 and –10, resulting in –5.
Examples & Analogies
Think of adding apples and oranges. If you have 7 apples (represented by 7x^2) and 5 apples (5) from another basket, and you later find 9 oranges (9x), you can express your total in terms of apples and oranges, combining only the apples appropriately.
Example of Adding Multiple Expressions
Chapter 3 of 5
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Chapter Content
Example 1: Add: 7xy + 5yz – 3zx, 4yz + 9zx – 4y, –2xy – 3zx + 5x. Solution:
7xy + 5yz –3zx + 4yz + 9zx – 4y + –2xy –3zx + 5x
Results in 5xy + 9yz + 3zx + 5x – 4y.
Detailed Explanation
In this example, we add three expressions by aligning like terms. For 7xy, 5yz, and –2xy, we group terms with the same variables. The result is a cohesive expression that represents the total combining all three individual expressions.
Examples & Analogies
Imagine you have three different boxes of fruit. One has 7 of one type, another has 5, and the third –2 (meaning you owe 2). You can combine these to find out how much fruit you have in total, just like the expressions are combined to find one total expression.
Subtracting Algebraic Expressions
Chapter 4 of 5
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Chapter Content
To subtract 5x^2 – 4y^2 + 6y – 3 from 7x^2 – 4xy + 8y^2 + 5x – 3y, we write:
7x^2 – 4xy + 8y^2 + 5x – 3y – (5x^2 – 4y^2 + 6y – 3)
This step allows us to see clearly which terms are being subtracted.
Detailed Explanation
Subtraction works similarly to addition, but you need to also change the signs of each term in the expression you are subtracting. Essentially, subtracting can be viewed as adding a negative. For example, subtracting 5x^2 translates to adding –5x^2.
Examples & Analogies
If you have $7 in one pocket and owe someone $5 (which is akin to subtracting it), you can think of it as putting your total into one equation where you start with positive funds and then deducting your debt to find your final balance.
Understanding Subtraction in Algebra
Chapter 5 of 5
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Chapter Content
Important: Subtracting a number is the same as adding its additive inverse. Thus, subtracting –3 is the same as adding +3. Similarly, subtracting 6y is like adding –6y; the same applies for –4y^2.
Detailed Explanation
This concept emphasizes that subtraction can often be simplified by changing the operation to addition. Understanding the additive inverse helps to clarify points of confusion, particularly when encountering negative numbers.
Examples & Analogies
Imagine you owe your friend $3 (subtracting $3 from your balance). When considering your money, it’s helpful to think of this as not just losing, but also as an incoming balance of $3 that you will need to account back to them from your future earnings.
Key Concepts
-
Like Terms: Terms that contain the same variables raised to the same power. For instance, in the expression
4xy + 5xy, both terms are like terms because they both contain the variablesxandy. -
Adding Expressions: To add multiple expressions, align like terms vertically to simplify the addition process. For example:
-
Adding
7x^2 - 4x + 5with9x - 10results in: -
7x^2 - 4x + 5
-
- 9x - 10
-
-
7x^2 + 5x - 5
-
Subtracting Expressions: Subtraction is equivalent to adding the additive inverse. Students will learn that subtracting a term means adding its opposite. For example, subtracting
5becomes adding-5. -
Significance
-
Understanding these operations allows students to manipulate algebraic expressions effectively, setting the stage for more complex algebraic concepts, including solving equations and functions.
Examples & Applications
Example of adding like terms: For the expressions 3x + 4x, the sum is 7x.
Example of subtracting: From 5x - 3, if we subtract 2x, we get 3x - 3.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Combine like terms, don't be shy, sums will soar high in the sky!
Stories
When Sandy and other kids (like terms) gather for a pizza dinner (addition), each kid ensures they bring the same amount of the same type of pizza!
Memory Tools
Always Keep in Mind (AKM): Align, Keep coefficients, Maintain signs.
Acronyms
A.A.S
Align
Add/Subtract
Simplify.
Flash Cards
Glossary
- Algebraic Expression
A mathematical statement that contains variables, constants, and operators.
- Like Terms
Terms in an expression that have the same variable raised to the same exponent.
- Coefficient
A numerical factor in a term of an algebraic expression.
- Additive Inverse
The opposite value of a number; it is the value that, when added to the original number, results in zero.
Reference links
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