Which Expressions Are Equivalent To $4\left(\frac{3}{4} Y-2+\frac{1}{2} Y\right)$? Choose ALL That Apply.A. 3 Y + 8 + 2 Y 3y + 8 + 2y 3 Y + 8 + 2 Y B. 5 Y − 8 5y - 8 5 Y − 8 C. − 3 Y -3y − 3 Y D. 4 Y − 8 4y - 8 4 Y − 8 E. 3 Y − 8 + 2 Y 3y - 8 + 2y 3 Y − 8 + 2 Y F. 5 Y + 8 5y + 8 5 Y + 8

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression $4\left(\frac{3}{4} y-2+\frac{1}{2} y\right)$. We will examine the different options provided and determine which ones are equivalent to the given expression.

Understanding the Given Expression

The given expression is $4\left(\frac{3}{4} y-2+\frac{1}{2} y\right)$. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Multiply the result by 4.

Step 1: Evaluate the Expressions Inside the Parentheses

The expression inside the parentheses is $\frac{3}{4} y-2+\frac{1}{2} y$. To evaluate this expression, we need to combine the like terms:

34y−2+12y=34y+12y−2\frac{3}{4} y-2+\frac{1}{2} y = \frac{3}{4} y + \frac{1}{2} y - 2

To combine the like terms, we need to find a common denominator, which is 4:

34y+12y=34y+24y=54y\frac{3}{4} y + \frac{1}{2} y = \frac{3}{4} y + \frac{2}{4} y = \frac{5}{4} y

So, the expression inside the parentheses simplifies to $\frac{5}{4} y - 2$.

Step 2: Multiply the Result by 4

Now that we have simplified the expression inside the parentheses, we can multiply the result by 4:

4(54y−2)=5y−84\left(\frac{5}{4} y - 2\right) = 5y - 8

Evaluating the Options

Now that we have simplified the given expression, we can evaluate the options provided:

  • A. 3y+8+2y3y + 8 + 2y is not equivalent to the given expression.
  • B. 5y−85y - 8 is equivalent to the given expression.
  • C. −3y-3y is not equivalent to the given expression.
  • D. 4y−84y - 8 is not equivalent to the given expression.
  • E. 3y−8+2y3y - 8 + 2y is not equivalent to the given expression.
  • F. 5y+85y + 8 is not equivalent to the given expression.

Conclusion

In conclusion, the only option that is equivalent to the given expression is B. 5y−85y - 8. This is because the given expression simplifies to 5y−85y - 8, which is the same as option B.

Tips and Tricks

When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS). This will ensure that you evaluate the expressions inside the parentheses correctly and multiply the result by the correct value.

Additionally, when combining like terms, it's crucial to find a common denominator to ensure that you are combining the terms correctly.

By following these tips and tricks, you will be able to simplify algebraic expressions with ease and confidence.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid:

  • Not following the order of operations (PEMDAS)
  • Not combining like terms correctly
  • Not finding a common denominator when combining like terms

By avoiding these common mistakes, you will be able to simplify algebraic expressions accurately and efficiently.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

  • Simplify the expression $3\left(2x - 4 + x\right)$
  • Simplify the expression $2\left(x + 3 - 2x\right)$
  • Simplify the expression $4\left(\frac{1}{2} x - 2 + \frac{3}{4} x\right)$

By practicing these problems, you will become more confident and proficient in simplifying algebraic expressions.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression $4\left(\frac{3}{4} y-2+\frac{1}{2} y\right)$. We also discussed the importance of following the order of operations (PEMDAS) and combining like terms correctly. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

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

Q: How do I combine like terms?

A: To combine like terms, you need to find a common denominator and 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=(2+3)x=5x2x + 3x = (2 + 3)x = 5x

Q: What is a common denominator?

A: A common denominator is a number that is a multiple of all the denominators in an expression. For example, if you have the expression $\frac{1}{2} x + \frac{1}{4} x$, the common denominator is 4. To combine the like terms, you can rewrite the expression as:

12x+14x=24x+14x=34x\frac{1}{2} x + \frac{1}{4} x = \frac{2}{4} x + \frac{1}{4} x = \frac{3}{4} x

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, you need to follow the order of operations (PEMDAS). First, evaluate any expressions inside the parentheses. Then, multiply the result by the coefficient outside the parentheses. For example, if you have the expression $4\left(2x - 3\right)$, you can simplify it as follows:

4(2x−3)=4(2x)−4(3)=8x−124\left(2x - 3\right) = 4(2x) - 4(3) = 8x - 12

Q: What is the difference between a coefficient and a constant?

A: A coefficient is a number that is multiplied by a variable. A constant is a number that is not multiplied by a variable. For example, in the expression $2x + 3$, the coefficient of x is 2, and the constant is 3.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to find a common denominator and add or subtract the fractions. For example, if you have the expression $\frac{1}{2} x + \frac{1}{4} x$, you can simplify it as follows:

12x+14x=24x+14x=34x\frac{1}{2} x + \frac{1}{4} x = \frac{2}{4} x + \frac{1}{4} x = \frac{3}{4} x

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill to master in mathematics. By following the order of operations (PEMDAS) and combining like terms correctly, you will be able to simplify expressions with ease and confidence. Remember to avoid common mistakes and practice regularly to become proficient in simplifying algebraic expressions.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

  • Simplify the expression $3\left(2x - 4 + x\right)$
  • Simplify the expression $2\left(x + 3 - 2x\right)$
  • Simplify the expression $4\left(\frac{1}{2} x - 2 + \frac{3}{4} x\right)$

By practicing these problems, you will become more confident and proficient in simplifying algebraic expressions.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid:

  • Not following the order of operations (PEMDAS)
  • Not combining like terms correctly
  • Not finding a common denominator when combining like terms

By avoiding these common mistakes, you will be able to simplify algebraic expressions accurately and efficiently.

Tips and Tricks

When simplifying algebraic expressions, here are some tips and tricks to keep in mind:

  • Always follow the order of operations (PEMDAS)
  • Combine like terms correctly
  • Find a common denominator when combining like terms
  • Practice regularly to become proficient in simplifying algebraic expressions

By following these tips and tricks, you will be able to simplify algebraic expressions with ease and confidence.