Evaluate The Following Expression Given $x = 3$ And $y = 5$:$\frac{3(y-x)^2+2}{2}$First, Plug In The Values In The Numerator.A. $5(3-2)^2+3$ B. $3(3-5)^2+2$ C. $3(5-3)^2+2$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will evaluate the expression 3(y−x)2+22\frac{3(y-x)^2+2}{2} given the values of x=3x = 3 and y=5y = 5. We will break down the expression into smaller parts, plug in the values, and simplify the expression step by step.

Step 1: Plug in the Values in the Numerator

To evaluate the expression, we need to start by plugging in the values of xx and yy into the numerator. We have three options:

A. 5(3−2)2+35(3-2)^2+3 B. 3(3−5)2+23(3-5)^2+2 C. 3(5−3)2+23(5-3)^2+2

Let's evaluate each option separately.

Option A: 5(3−2)2+35(3-2)^2+3

To evaluate this option, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 3−2=13-2 = 1
  2. Raise the result to the power of 2: 12=11^2 = 1
  3. Multiply the result by 5: 5×1=55 \times 1 = 5
  4. Add 3 to the result: 5+3=85 + 3 = 8

So, the value of option A is 8.

Option B: 3(3−5)2+23(3-5)^2+2

To evaluate this option, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 3−5=−23-5 = -2
  2. Raise the result to the power of 2: (−2)2=4(-2)^2 = 4
  3. Multiply the result by 3: 3×4=123 \times 4 = 12
  4. Add 2 to the result: 12+2=1412 + 2 = 14

So, the value of option B is 14.

Option C: 3(5−3)2+23(5-3)^2+2

To evaluate this option, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 5−3=25-3 = 2
  2. Raise the result to the power of 2: 22=42^2 = 4
  3. Multiply the result by 3: 3×4=123 \times 4 = 12
  4. Add 2 to the result: 12+2=1412 + 2 = 14

So, the value of option C is also 14.

Discussion

Now that we have evaluated all three options, we can see that options B and C have the same value, which is 14. This means that the correct option is B or C.

However, we need to go back to the original expression and plug in the values of xx and yy into the numerator. We have:

3(y−x)2+22\frac{3(y-x)^2+2}{2}

Given x=3x = 3 and y=5y = 5, we can plug in the values as follows:

3(5−3)2+22\frac{3(5-3)^2+2}{2}

This is option C, which we already evaluated and found to be 14.

Conclusion

In conclusion, the correct value of the expression 3(y−x)2+22\frac{3(y-x)^2+2}{2} given x=3x = 3 and y=5y = 5 is 14.

Final Answer

The final answer is 14\boxed{14}.

Additional Tips and Resources

  • To evaluate algebraic expressions, always follow the order of operations (PEMDAS).
  • Make sure to plug in the values correctly and simplify the expression step by step.
  • Practice evaluating algebraic expressions with different values and variables to become more confident and proficient.

References

Introduction

In our previous article, we evaluated the expression 3(y−x)2+22\frac{3(y-x)^2+2}{2} given the values of x=3x = 3 and y=5y = 5. We broke down the expression into smaller parts, plugged in the values, and simplified the expression step by step. In this article, we will answer some frequently asked questions (FAQs) related to evaluating algebraic expressions.

Q&A

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 we have multiple operations in an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
  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 evaluate an expression with multiple variables?

A: To evaluate an expression with multiple variables, you need to plug in the values of each variable into the expression. For example, if we have the expression 2x+3y2x + 3y and we know that x=4x = 4 and y=5y = 5, we can plug in the values as follows:

2(4)+3(5)2(4) + 3(5)

First, evaluate the expressions inside the parentheses:

8+158 + 15

Then, add the results:

2323

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

A: A variable is a symbol that represents a value that can change. For example, xx is a variable. A constant is a value that does not change. For example, 55 is a constant.

Q: How do I simplify an expression with fractions?

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

  1. Find the least common multiple (LCM) of the denominators.
  2. Multiply each fraction by the LCM.
  3. Simplify the resulting expression.

For example, if we have the expression 12+13\frac{1}{2} + \frac{1}{3}, we can follow these steps:

  1. Find the LCM of 2 and 3, which is 6.
  2. Multiply each fraction by 6:

62+63\frac{6}{2} + \frac{6}{3}

  1. Simplify the resulting expression:

3+23 + 2

55

Q: Can I use a calculator to evaluate an expression?

A: Yes, you can use a calculator to evaluate an expression. However, make sure to follow the order of operations (PEMDAS) and simplify the expression step by step.

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you need to follow these steps:

  1. Evaluate any exponential expressions first (e.g., 2^3).
  2. Then, evaluate any multiplication and division operations from left to right.
  3. Finally, evaluate any addition and subtraction operations from left to right.

For example, if we have the expression 23×4+52^3 \times 4 + 5, we can follow these steps:

  1. Evaluate the exponential expression:

8×4+58 \times 4 + 5

  1. Multiply 8 and 4:

32+532 + 5

  1. Add 5 to the result:

3737

Conclusion

Evaluating algebraic expressions is a crucial skill for students and professionals alike. By following the order of operations (PEMDAS) and simplifying the expression step by step, you can evaluate even the most complex expressions. Remember to plug in the values of each variable, simplify fractions, and use a calculator only when necessary. With practice and patience, you will become proficient in evaluating algebraic expressions.

Additional Tips and Resources