Simplify The Given Algebraic Expression.${ 7(5y - 2) - (7y + 2) }$ { 7(5y - 2) - (7y + 2) = \square \}

Understanding the Algebraic Expression

The given algebraic expression is $7(5y - 2) - (7y + 2)$. This expression involves the use of parentheses, which are essential in algebra to group and order operations. The expression consists of two main parts: the first part is $7(5y - 2)$, and the second part is $-(7y + 2)$. Our goal is to simplify this expression by applying the rules of algebra.

Step 1: Distribute the 7 to the Terms Inside the Parentheses

To simplify the expression, we need to start by distributing the 7 to the terms inside the parentheses. This means multiplying the 7 by each term inside the parentheses. We can write this as:

$$7(5y - 2) = 7(5y) - 7(2)$$

Using the distributive property, we can simplify this further:

$$7(5y) = 35y$$

$$-7(2) = -14$$

So, the expression becomes:

$$35y - 14 - (7y + 2)$$

Step 2: Distribute the Negative Sign to the Terms Inside the Parentheses

Now, we need to distribute the negative sign to the terms inside the parentheses. This means multiplying the negative sign by each term inside the parentheses. We can write this as:

$$-(7y + 2) = -7y - 2$$

Now, we can combine the two parts of the expression:

$$35y - 14 - 7y - 2$$

Step 3: Combine Like Terms

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable y: $35y$ and $-7y$. We can combine these terms by adding their coefficients:

$$35y - 7y = 28y$$

Now, we can combine the constant terms:

$$-14 - 2 = -16$$

So, the simplified expression is:

$$28y - 16$$

Conclusion

In this article, we simplified the given algebraic expression $7(5y - 2) - (7y + 2)$ by applying the rules of algebra. We started by distributing the 7 to the terms inside the parentheses, then distributed the negative sign to the terms inside the parentheses, and finally combined like terms. The simplified expression is $28y - 16$.

Tips and Tricks

• When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS).
• Use parentheses to group and order operations.
• Distribute coefficients to the terms inside the parentheses.
• Combine like terms to simplify the expression.

Common Mistakes

• Failing to distribute coefficients to the terms inside the parentheses.
• Failing to combine like terms.
• Not following the order of operations (PEMDAS).

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics and has many real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In economics, algebraic expressions are used to model economic systems. In computer science, algebraic expressions are used to write algorithms and programs.

Practice Problems

1. Simplify the algebraic expression $3(2x + 5) - (x - 2)$.
2. Simplify the algebraic expression $2(3y - 4) + (y + 2)$.
3. Simplify the algebraic expression $5(2x + 3) - (4x - 2)$.

Answer Key

1. $6x + 13$
2. $7y - 6$
3. $10x + 13$

References

• Algebraic Expression
• Distributive Property
• Like Terms

Conclusion

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute a coefficient to the terms inside a set of parentheses. It states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow these steps:

1. Distribute the coefficients to the terms inside the parentheses.
2. Combine like terms.
3. Simplify the expression by removing any unnecessary parentheses or brackets.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression 2x + 4x, you can combine the like terms by adding the coefficients:

2x + 4x = 6x

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I use the order of operations to simplify an expression?

A: To use the order of operations to simplify an expression, you need to follow these steps:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents.
3. Perform any multiplication and division operations from left to right.
4. Perform any addition and subtraction operations from left to right.

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

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

• Failing to distribute coefficients to the terms inside the parentheses.
• Failing to combine like terms.
• Not following the order of operations (PEMDAS).

Q: How do I check my work when simplifying an algebraic expression?

A: To check your work when simplifying an algebraic expression, you need to follow these steps:

1. Simplify the expression using the steps above.
2. Check that you have distributed the coefficients correctly.
3. Check that you have combined like terms correctly.
4. Check that you have followed the order of operations correctly.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects.
• Economics: Algebraic expressions are used to model economic systems.
• Computer Science: Algebraic expressions are used to write algorithms and programs.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

• Working through practice problems.
• Using online resources and tools.
• Asking a teacher or tutor for help.

Q: What are some resources for learning more about simplifying algebraic expressions?

A: Some resources for learning more about simplifying algebraic expressions include:

• Online tutorials and videos.
• Textbooks and workbooks.
• Online communities and forums.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics that has many real-world applications. By following the rules of algebra and applying the distributive property, we can simplify complex expressions and make them easier to understand. With practice and patience, anyone can master the art of simplifying algebraic expressions.