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Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Apply the Properties of Operations to Generate Equivalent Expressions 🧮

Imagine you are playing a building game, like Minecraft or Roblox. You can build a house using different sets of blocks, but in the end, the house still takes up the same space and looks the same from the outside. That is what we do in algebra when we write equivalent expressions—they may look different, but they have the same value for every number you put in. 🎯

In this lesson, you will learn how to use the properties of operations (like the distributive property) to change expressions into different but equal forms. This is a key skill in algebra and will help you with equations, word problems, and even real-life situations like shopping and splitting costs with friends.

The main idea: use properties of operations to rewrite expressions, such as turning \(3(2 + x)\) into \(6 + 3x\), or turning \(y + y + y\) into \(3y\).

The idea of “same value, different look” is shown clearly in [Figure 1], where blocks are grouped in different ways but represent the same total.

What Are Equivalent Expressions?

Equivalent expressions are expressions that have the same value for every value of the variable, even if they look different.

Examples:

• \(3(2 + x)\) and \(6 + 3x\)
• \(y + y + y\) and \(3y\)
• \(24x + 18y\) and \(6(4x + 3y)\)

To check if two expressions are equivalent, you can plug in the same value for the variable.

Example check: Are \(y + y + y\) and \(3y\) equivalent?

• Let \(y = 4\).
• \(y + y + y = 4 + 4 + 4 = 12\).
• \(3y = 3 \cdot 4 = 12\).

They give the same result, so they are equivalent.

The Big Three: Properties of Operations

We mainly use three properties to generate equivalent expressions:

1. Commutative Property
2. Associative Property
3. Distributive Property

You have seen these with numbers in earlier grades. Now we extend them to algebraic expressions that include variables like \(x\) and \(y\).

1. Commutative Property (Order Doesn’t Matter for + and ×)

The commutative property tells us we can change the order when we add or multiply, and the value stays the same.

• Addition: \(a + b = b + a\)
• Multiplication: \(ab = ba\)

Examples with numbers:

• \(3 + 5 = 5 + 3\)
• \(2 \cdot 7 = 7 \cdot 2\)

Examples with variables:

• \(x + 4 = 4 + x\)
• \(3y = y \cdot 3\)
• \(2x + 5 = 5 + 2x\)

These are all equivalent expressions because only the order changed.

2. Associative Property (Grouping Doesn’t Matter for + and ×)

The associative property tells us we can change the grouping when adding or multiplying, and the value stays the same.

• Addition: \((a + b) + c = a + (b + c)\)
• Multiplication: \((ab)c = a(bc)\)

Examples with numbers:

• \((2 + 3) + 4 = 2 + (3 + 4)\)
• \((2 \cdot 5) \cdot 3 = 2 \cdot (5 \cdot 3)\)

Examples with variables:

• \((x + 2) + 5 = x + (2 + 5)\)
• \((2x \cdot 3)y = 2x(3y)\)

We are not changing the numbers or variables, just how they are grouped.

3. Distributive Property (The Main Star 🌟)

The distributive property connects multiplication and addition (or subtraction). It says:

\[a(b + c) = ab + ac\]

We “distribute” the outside factor \(a\) to each term inside the parentheses.

Examples with numbers:

• \(3(2 + 4) = 3 \cdot 2 + 3 \cdot 4 = 6 + 12 = 18\)
• Check: \(3(2 + 4) = 3 \cdot 6 = 18\). Same result.

Examples with variables:

• \(3(2 + x) = 3 \cdot 2 + 3 \cdot x = 6 + 3x\)
• \(5(x + 7) = 5x + 35\)
• \(2(a + b + c) = 2a + 2b + 2c\)

We can also use it in reverse to factor, which means putting something back into parentheses:

\[ab + ac = a(b + c)\]

For example, \(24x + 18y = 6(4x + 3y)\) because \(6 \cdot 4x = 24x\) and \(6 \cdot 3y = 18y\).

The steps of distributing and factoring are shown clearly in [Figure 2], using arrows from the outside factor to each term inside the parentheses.

Using Properties to Generate Equivalent Expressions

Now we put these properties to work to create equivalent expressions.

Example Type 1: Distributing (Expanding Expressions)

Goal: Remove parentheses by multiplying.

Example A: \(3(2 + x)\)

Step-by-step solution

1. Identify the outside factor: \(3\).
2. Identify the terms inside the parentheses: \(2\) and \(x\).
3. Multiply 3 by each term:
   • \(3 \cdot 2 = 6\)
   • \(3 \cdot x = 3x\)
4. Write the sum: \(6 + 3x\).

So \(3(2 + x)\) and \(6 + 3x\) are equivalent expressions.

Example B: \(4(x + 5)\)

1. Outside factor: \(4\).
2. Inside terms: \(x\) and \(5\).
3. Multiply:
   • \(4 \cdot x = 4x\)
   • \(4 \cdot 5 = 20\)
4. Result: \(4x + 20\).

So \(4(x + 5) = 4x + 20\).

Example C: \(2(3y + 4)\)

1. Outside factor: \(2\).
2. Inside terms: \(3y\) and \(4\).
3. Multiply:
   • \(2 \cdot 3y = 6y\)
   • \(2 \cdot 4 = 8\)
4. Result: \(6y + 8\).

Example Type 2: Factoring (Using the Distributive Property Backwards)

Goal: Put an expression back into parentheses with a common factor outside.

Example D: \(24x + 18y\)

Step-by-step solution

1. Find the greatest common factor (GCF) of 24 and 18. The GCF is 6.
2. Both terms have variables \(x\) or \(y\), but there is no common variable, so we only take 6 as the common factor.
3. Divide each term by 6 to see what stays inside the parentheses:
   • \(24x \div 6 = 4x\)
   • \(18y \div 6 = 3y\)
4. Write the factored form: \(6(4x + 3y)\).

So \(24x + 18y\) and \(6(4x + 3y)\) are equivalent.

Example E: \(12x + 8\)

1. GCF of 12 and 8 is 4.
2. Factor out 4: \(12x = 4 \cdot 3x\), \(8 = 4 \cdot 2\).
3. Write: \(12x + 8 = 4(3x + 2)\).

Example Type 3: Combining Like Terms Using Properties

We say terms are like terms if they have the same variable raised to the same power.

• Like terms: \(2x\) and \(5x\), \(3y\) and \(y\).
• Not like terms: \(x\) and \(y\), \(x\) and \(x^2\).

The property behind combining like terms is actually the distributive property plus the commutative and associative properties.

Example F: \(y + y + y\)

Step-by-step solution

1. There are three \(y\) terms.
2. Use the associative and commutative properties to think of this as \((1y + 1y + 1y)\).
3. Group the coefficients (the numbers in front of the variable): \((1 + 1 + 1)y\).
4. Add the coefficients: \(1 + 1 + 1 = 3\).
5. Result: \(3y\).

So \(y + y + y\) is equivalent to \(3y\).

This process is shown visually in [Figure 3], where three \(y\)-tiles are grouped and then replaced by a single block labeled \(3y\).

Solved Examples (Step-by-Step) ✅

Here are three fully worked examples to practice seeing how properties create equivalent expressions.

Solved Example 1: Expand Using Distributive Property

Problem: Rewrite \(5(2x + 3)\) as an equivalent expression without parentheses.

Solution:

1. Identify the outside factor: \(5\).
2. Identify inside terms: \(2x\) and \(3\).
3. Distribute 5 to each term:
   • \(5 \cdot 2x = 10x\)
   • \(5 \cdot 3 = 15\)
4. Write the sum: \(10x + 15\).

Answer: \(5(2x + 3) = 10x + 15\).

Solved Example 2: Factor Using Distributive Property

Problem: Write \(14x + 21\) as a product using the distributive property.

Solution:

1. Find the GCF of 14 and 21. The GCF is 7.
2. Factor 7 out of each term:
   • \(14x = 7 \cdot 2x\)
   • \(21 = 7 \cdot 3\)
3. Use the distributive property backwards:
   • \(14x + 21 = 7(2x + 3)\)

Answer: \(14x + 21 = 7(2x + 3)\).

Solved Example 3: Combine Like Terms

Problem: Simplify \(2y + 5 + 3y\) to an equivalent expression with as few terms as possible.

Solution:

1. Reorder using the commutative property so like terms are together:
   • \(2y + 3y + 5\)
2. Combine \(2y\) and \(3y\). Think of it as \((2 + 3)y\):
   • \(2 + 3 = 5\)
   • So \(2y + 3y = 5y\)
3. Keep the constant term 5:
   • Expression is \(5y + 5\)

Answer: \(2y + 5 + 3y = 5y + 5\).

More Practice-Style Examples (Without Full Steps)

• \(6(3 + x) = 18 + 6x\)
• \(8x + 12 = 4(2x + 3)\)
• \(a + a + a + a = 4a\)
• \(2(4y + 1) = 8y + 2\)
• \(9m + 6n = 3(3m + 2n)\)

Real-World Applications 🌍

These properties are not just for math class. They show up in everyday situations.

1. Shopping and Discounts

Suppose you buy 3 T-shirts, each costing \(x\) dollars, and each has an extra decoration that costs 2 dollars. The total cost is:

\[3(x + 2)\]

This means “3 groups of (shirt price + decoration).” Using the distributive property:

\[3(x + 2) = 3x + 6\]

Now you see the same situation as “total shirt cost” plus “total decoration cost.” Both expressions are equivalent and describe the same purchase. 💡

2. Sharing Snacks

Imagine you have 4 bags, and each bag has \(2 + y\) cookies (2 chocolate cookies and \(y\) sugar cookies). The total number of cookies is:

\[4(2 + y)\]

Using the distributive property:

\[4(2 + y) = 8 + 4y\]

This tells you there are 8 chocolate cookies and \(4y\) sugar cookies in total.

3. Video Game Rewards 🎮

Say you play 5 rounds of a game. In each round you earn 10 coins plus \(p\) bonus points. The total reward is:

\[5(10 + p)\]

Using the distributive property:

\[5(10 + p) = 50 + 5p\]

So you always get 50 coins plus \(5p\) bonus points overall. Whether you think in groups \(5(10 + p)\) or as a total \(50 + 5p\), it is the same amount.

Why This Skill Is Powerful ⚡

Using properties of operations to generate equivalent expressions helps you:

• Simplify long or messy expressions.
• Factor expressions to make them easier to work with later (for example, when solving equations).
• Understand the structure of algebra, like how multiplication and addition connect.
• Model real-life situations with expressions that you can transform into more useful forms.

Key Ideas to Remember 🎉

• Equivalent expressions have the same value for all values of the variable, even if they look different.
• Commutative property: You can change the order for addition and multiplication.
  • \(a + b = b + a\), \(ab = ba\).
• Associative property: You can change the grouping for addition and multiplication.
  • \((a + b) + c = a + (b + c)\).
• Distributive property: Multiply a number by a sum or difference by distributing it to each term inside.
  • \(a(b + c) = ab + ac\).
  • Works in reverse for factoring: \(ab + ac = a(b + c)\).
• Expressions like \(y + y + y\) use these properties to become \(3y\).
• You use these ideas every time you simplify expressions, factor them, or model real-life situations with algebra.