Simplify The Expression: ${ 5(3p + 3) - (3p + 10)\$}

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Introduction

In this article, we will simplify the given algebraic expression: $5(3p + 3) - (3p + 10)$. This expression involves the distributive property, which is a fundamental concept in algebra. We will use the distributive property to expand the terms and then combine like terms to simplify the expression.

Understanding the Distributive Property

The distributive property is a mathematical concept that allows us to expand the terms in an algebraic expression. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

This property can be applied to any algebraic expression that involves the multiplication of a term by a sum of terms.

Expanding the Expression

To simplify the given expression, we will first expand the terms using the distributive property. We will start by expanding the first term, $5(3p + 3)$.

$$5(3p + 3) = 5 \cdot 3p + 5 \cdot 3$$

Using the distributive property, we can rewrite this expression as:

$$15p + 15$$

Now, we will expand the second term, $(3p + 10)$.

$$(3p + 10) = 3p + 10$$

This expression is already expanded, so we can move on to the next step.

Combining Like Terms

Now that we have expanded both terms, we can combine like terms to simplify the expression. We will start by combining the like terms in the first term, $15p + 15$.

$$15p + 15 = 15p + 15$$

There are no like terms in this expression, so we can move on to the next step.

Next, we will combine the like terms in the second term, $3p + 10$.

$$3p + 10 = 3p + 10$$

Again, there are no like terms in this expression, so we can move on to the next step.

Simplifying the Expression

Now that we have expanded and combined like terms, we can simplify the expression by subtracting the second term from the first term.

$$5(3p + 3) - (3p + 10) = (15p + 15) - (3p + 10)$$

Using the distributive property, we can rewrite this expression as:

$$15p + 15 - 3p - 10$$

Now, we can combine like terms to simplify the expression.

$$15p - 3p + 15 - 10$$

Combining like terms, we get:

$$12p + 5$$

Therefore, the simplified expression is:

$$12p + 5$$

Conclusion

In this article, we simplified the given algebraic expression: $5(3p + 3) - (3p + 10)$. We used the distributive property to expand the terms and then combined like terms to simplify the expression. The final simplified expression is $12p + 5$. This expression can be used in a variety of mathematical contexts, such as solving equations and inequalities.

Real-World Applications

The distributive property is a fundamental concept in algebra that has many real-world applications. For example, in business, the distributive property can be used to calculate the total cost of a product by multiplying the cost per unit by the number of units sold. In science, the distributive property can be used to calculate the total energy of a system by multiplying the energy per unit by the number of units.

Tips and Tricks

When simplifying algebraic expressions, it is essential to use the distributive property to expand the terms. This will help you to identify like terms and combine them to simplify the expression. Additionally, when combining like terms, make sure to combine the coefficients of the like terms.

Common Mistakes

When simplifying algebraic expressions, there are several common mistakes that can be made. One common mistake is to forget to use the distributive property to expand the terms. Another common mistake is to combine like terms incorrectly. To avoid these mistakes, make sure to use the distributive property to expand the terms and combine like terms carefully.

Final Thoughts

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Introduction

In our previous article, we simplified the given algebraic expression: $5(3p + 3) - (3p + 10)$. We used the distributive property to expand the terms and then combined like terms to simplify the expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to expand the terms in an algebraic expression. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

Q: How do I use the distributive property to simplify an algebraic expression?

A: To simplify an algebraic expression using the distributive property, you need to expand the terms by multiplying each term by the other terms. For example, if you have the expression $5(3p + 3)$, you would multiply the $5$ by each term inside the parentheses:

$$5(3p + 3) = 5 \cdot 3p + 5 \cdot 3$$

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $3p$ and $5p$ are like terms because they both have the variable $p$ raised to the power of $1$. Like terms can be combined by adding or subtracting their coefficients.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, if you have the expression $3p + 5p$, you would combine the like terms by adding their coefficients:

$$3p + 5p = 8p$$

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an algebraic expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression with multiple terms?

A: To simplify an algebraic expression with multiple terms, you need to use the distributive property to expand the terms and then combine like terms. For example, if you have the expression $5(3p + 3) - (3p + 10)$, you would first expand the terms using the distributive property:

$$5(3p + 3) = 15p + 15$$

$$(3p + 10) = 3p + 10$$

Then, you would combine like terms:

$$15p + 15 - 3p - 10 = 12p + 5$$

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Forgetting to use the distributive property to expand the terms.
• Combining like terms incorrectly.
• Not following the order of operations.
• Not simplifying the expression completely.

Conclusion

In this article, we answered some frequently asked questions about simplifying algebraic expressions. We covered topics such as the distributive property, like terms, combining like terms, the order of operations, and common mistakes to avoid. By mastering these concepts, you can simplify complex algebraic expressions and solve mathematical problems with ease.

Real-World Applications

The concepts covered in this article have many real-world applications. For example, in business, the distributive property can be used to calculate the total cost of a product by multiplying the cost per unit by the number of units sold. In science, the distributive property can be used to calculate the total energy of a system by multiplying the energy per unit by the number of units.

Tips and Tricks

When simplifying algebraic expressions, it is essential to use the distributive property to expand the terms. This will help you to identify like terms and combine them to simplify the expression. Additionally, when combining like terms, make sure to combine the coefficients of the like terms.

Common Mistakes

When simplifying algebraic expressions, there are several common mistakes that can be made. One common mistake is to forget to use the distributive property to expand the terms. Another common mistake is to combine like terms incorrectly. To avoid these mistakes, make sure to use the distributive property to expand the terms and combine like terms carefully.

Final Thoughts

In conclusion, simplifying algebraic expressions is an essential skill in mathematics. By using the distributive property to expand the terms and combining like terms, we can simplify complex expressions and solve mathematical problems. The concepts covered in this article have many real-world applications, and by mastering these concepts, you can solve a wide range of mathematical problems and apply mathematical concepts to real-world situations.