Expand The Expression:${ \begin{aligned} 0.25\left(\frac{2}{3} Y - 5 + \frac{1}{3} Y - 7 - 9y\right) & = 0.25(-8y - 12) \ & = \square Y - \square \end{aligned} }$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on expanding and simplifying a given algebraic expression, step by step. We will use the expression $0.25\left(\frac{2}{3} y - 5 + \frac{1}{3} y - 7 - 9y\right)$ as an example and break it down into manageable parts.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at it. The expression is a combination of fractions, decimals, and variables. We can see that it involves the variable $y$, which is multiplied by various coefficients. Our goal is to simplify this expression and rewrite it in a more manageable form.

Step 1: Distribute the Coefficient

The first step in simplifying the expression is to distribute the coefficient $0.25$ to each term inside the parentheses. This means that we need to multiply $0.25$ by each of the terms $\frac{2}{3} y$, $-5$, $\frac{1}{3} y$, $-7$, and $-9y$.

0.25\left(\frac{2}{3} y - 5 + \frac{1}{3} y - 7 - 9y\right) = 0.25\left(\frac{2}{3} y\right) - 0.25(5) + 0.25\left(\frac{1}{3} y\right) - 0.25(7) - 0.25(9y)

Step 2: Simplify the Fractions

Now that we have distributed the coefficient, we can simplify the fractions. We can start by simplifying the fractions $\frac{2}{3} y$ and $\frac{1}{3} y$. To do this, we need to multiply each fraction by the reciprocal of its denominator.

0.25\left(\frac{2}{3} y\right) = 0.25\left(\frac{2}{3}\right)y = \frac{0.25 \times 2}{3}y = \frac{0.5}{3}y
0.25\left(\frac{1}{3} y\right) = 0.25\left(\frac{1}{3}\right)y = \frac{0.25 \times 1}{3}y = \frac{0.25}{3}y

Step 3: Combine Like Terms

Now that we have simplified the fractions, we can combine like terms. We can start by combining the terms $\frac{0.5}{3}y$ and $\frac{0.25}{3}y$. To do this, we need to add the coefficients of the two terms.

\frac{0.5}{3}y + \frac{0.25}{3}y = \left(\frac{0.5}{3} + \frac{0.25}{3}\right)y = \frac{0.75}{3}y

Step 4: Simplify the Expression

Now that we have combined like terms, we can simplify the expression. We can start by multiplying the coefficient $0.25$ by the term $-8y$.

0.25(-8y) = -2y

Step 5: Rewrite the Expression

Finally, we can rewrite the expression in a more manageable form. We can start by combining the terms $-2y$ and $-12$.

-2y - 12

Conclusion

In this article, we have simplified the algebraic expression $0.25\left(\frac{2}{3} y - 5 + \frac{1}{3} y - 7 - 9y\right)$ step by step. We have distributed the coefficient, simplified the fractions, combined like terms, and rewritten the expression in a more manageable form. By following these steps, we can simplify any algebraic expression and rewrite it in a more manageable form.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid. Here are a few:

• Not distributing the coefficient: Failing to distribute the coefficient to each term inside the parentheses can lead to incorrect simplifications.
• Not simplifying fractions: Failing to simplify fractions can lead to incorrect simplifications.
• Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
• Not rewriting the expression: Failing to rewrite the expression in a more manageable form can make it difficult to work with.

Tips and Tricks

Here are a few tips and tricks to help you simplify algebraic expressions:

• Use a systematic approach: Use a systematic approach to simplify algebraic expressions, such as distributing the coefficient, simplifying fractions, combining like terms, and rewriting the expression.
• Use algebraic properties: Use algebraic properties, such as the distributive property and the commutative property, to simplify algebraic expressions.
• Check your work: Check your work to ensure that you have simplified the expression correctly.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Here are a few:

• Science and engineering: Simplifying algebraic expressions is essential in science and engineering, where complex equations need to be solved to model real-world phenomena.
• Economics: Simplifying algebraic expressions is essential in economics, where complex equations need to be solved to model economic systems.
• Computer science: Simplifying algebraic expressions is essential in computer science, where complex algorithms need to be implemented to solve real-world problems.

Conclusion

Introduction

Simplifying algebraic expressions is an essential skill for any math enthusiast. In our previous article, we provided a step-by-step guide on how to simplify algebraic expressions. In this article, we will answer some frequently asked questions (FAQs) about simplifying algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. Algebraic expressions are used to represent mathematical relationships and can be simplified to make them easier to work with.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is important because it makes them easier to work with and understand. Simplified expressions can be used to solve equations, graph functions, and model real-world phenomena.

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

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

• Not distributing the coefficient to each term inside the parentheses
• Not simplifying fractions
• Not combining like terms
• Not rewriting the expression in a more manageable form

Q: How do I distribute the coefficient to each term inside the parentheses?

A: To distribute the coefficient to each term inside the parentheses, you need to multiply the coefficient by each term. For example, if you have the expression $2(x + 3)$, you would distribute the coefficient 2 to each term inside the parentheses by multiplying 2 by x and 2 by 3.

Q: How do I simplify fractions?

A: To simplify fractions, you need to multiply the numerator and denominator by the reciprocal of the denominator. For example, if you have the fraction $\frac{2}{3}$, you can simplify it by multiplying the numerator and denominator by 3.

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 $2x + 3x$, you can combine the like terms by adding the coefficients: $2x + 3x = 5x$.

Q: How do I rewrite the expression in a more manageable form?

A: To rewrite the expression in a more manageable form, you need to simplify the expression by combining like terms and eliminating any unnecessary terms. For example, if you have the expression $2x + 3x - 2x$, you can rewrite it in a more manageable form by combining the like terms: $2x + 3x - 2x = 3x$.

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

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

• Science and engineering: Simplifying algebraic expressions is essential in science and engineering, where complex equations need to be solved to model real-world phenomena.
• Economics: Simplifying algebraic expressions is essential in economics, where complex equations need to be solved to model economic systems.
• Computer science: Simplifying algebraic expressions is essential in computer science, where complex algorithms need to be implemented to solve real-world problems.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems and exercises. You can also use online resources, such as algebraic expression simplifiers and calculators, to help you practice.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following a systematic approach, using algebraic properties, and checking your work, you can simplify any algebraic expression and rewrite it in a more manageable form. Remember to avoid common mistakes and use tips and tricks to help you simplify algebraic expressions. With practice and patience, you can become proficient in simplifying algebraic expressions and apply them to real-world problems.

Additional Resources

If you need additional help or resources to simplify algebraic expressions, here are some additional resources you can use:

• Algebraic expression simplifiers and calculators
• Online algebraic expression simplification tools
• Algebraic expression simplification worksheets and exercises
• Algebraic expression simplification videos and tutorials