Evaluating The Expression 2y^2 + (10y)/x Given X=5 And Y=-3

Let's dive into this mathematical problem where we need to evaluate the expression 2y^2 + (10y)/x given that x = 5 and y = -3. This type of problem is a classic example of substituting values into an algebraic expression, a fundamental skill in algebra. Guys, this is something you'll encounter often, so let's break it down step-by-step to make sure you've got it. We'll go through the substitution process, simplify the expression, and arrive at the correct answer. Understanding the order of operations (PEMDAS/BODMAS) is crucial here, so we'll keep that in mind as we work through the problem.

Step-by-Step Solution

1. Substitution

The first thing we need to do is substitute the given values of x and y into the expression. We have x = 5 and y = -3, and our expression is 2y^2 + (10y)/x. Replacing x and y with their respective values, we get:

2(-3)^2 + (10(-3))/5

This step is all about careful replacement. Make sure you're putting the values in the right spots. A small mistake here can throw off your entire calculation, so double-check your substitution before moving on. It's like making sure all the ingredients are correct before you start baking a cake – you don't want to end up with something unexpected!

2. Exponents

Now that we've substituted the values, we need to simplify the expression following the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). The first operation we need to tackle is the exponent. We have (-3)^2, which means -3 multiplied by itself:

(-3)^2 = (-3) * (-3) = 9

So, our expression now looks like this:

2(9) + (10(-3))/5

Dealing with exponents first is a key step. It ensures we're on the right track to simplifying the expression correctly. Exponents can sometimes be tricky, especially with negative numbers, so take your time and make sure you've got the sign right. A negative number squared becomes positive, and that's what happened here.

3. Multiplication

Next up, we have multiplication. We've got two multiplication operations in our expression: 2(9) and 10(-3). Let's take them one at a time:

• 2(9) = 18
• 10(-3) = -30

Now, our expression looks like this:

18 + (-30)/5

Multiplication is pretty straightforward, but again, watch out for those signs! Multiplying a positive number by a negative number gives you a negative result, and that's why 10 multiplied by -3 is -30. Keeping track of the signs is crucial for getting the correct final answer. It's like following a recipe – you need to measure the ingredients accurately to get the dish just right.

4. Division

After multiplication, we move on to division. We have (-30)/5, which is -30 divided by 5:

(-30)/5 = -6

Our expression is now simplified to:

18 + (-6)

Division is the inverse operation of multiplication, and it's just as important to handle it carefully. Remember, dividing a negative number by a positive number results in a negative number. This step brings us closer to the final answer, and we're almost there!

5. Addition

Finally, we have addition. We need to add 18 and -6:

18 + (-6) = 18 - 6 = 12

So, the value of the expression 2y^2 + (10y)/x when x = 5 and y = -3 is 12.

Final Answer

Therefore, the correct answer is:

C. 12

Common Mistakes to Avoid

When tackling problems like this, there are a few common pitfalls that students often encounter. Let's highlight these so you can steer clear of them:

• Incorrect Substitution: As we mentioned earlier, a mistake in the initial substitution can derail the entire problem. Double-check that you've placed the values of x and y correctly in the expression.
• Order of Operations: Forgetting the order of operations (PEMDAS/BODMAS) is a classic mistake. Make sure you perform exponents before multiplication and division, and multiplication and division before addition and subtraction.
• Sign Errors: Dealing with negative numbers can be tricky. Pay close attention to the signs when squaring, multiplying, and dividing. A small sign error can lead to a wrong answer.
• Arithmetic Errors: Simple calculation mistakes can happen, especially under pressure. Take your time and double-check your arithmetic to avoid these errors.

By being aware of these common mistakes, you can increase your chances of getting the correct answer and build confidence in your algebraic skills.

Practice Problems

To solidify your understanding of evaluating algebraic expressions, let's try a couple of practice problems. These will give you a chance to apply the steps we've discussed and identify any areas where you might need more practice.

Practice Problem 1

Evaluate the expression 3a^2 - 5b when a = -2 and b = 4.

Practice Problem 2

Find the value of (4x + 2y) / z if x = 3, y = -1, and z = 2.

Work through these problems on your own, following the steps we've outlined. Check your answers carefully, and don't hesitate to review the solution if you get stuck. Practice makes perfect, and the more you work with algebraic expressions, the more comfortable you'll become with them.

Conclusion

Evaluating algebraic expressions is a fundamental skill in mathematics. By understanding the order of operations and practicing careful substitution, you can confidently tackle these types of problems. Remember, guys, the key is to break the problem down into smaller, manageable steps and pay attention to detail. Keep practicing, and you'll become a pro at evaluating expressions in no time! We've seen how to substitute values, handle exponents, perform multiplication and division, and finally, add the results to get to our answer. Keep an eye out for those pesky signs and common mistakes, and you'll be golden. Now, go forth and conquer those algebraic expressions!