Evaluate (2y+7)/(-4y-10) When Y=-3

by Kenji Nakamura 35 views

Hey guys! Let's break down this math problem together. We've got an expression to evaluate, and it involves a fraction with the variable 'y'. The key here is that we know exactly what 'y' is equal to: -3. So, our mission is to substitute -3 wherever we see 'y' in the expression and then simplify things to get our final answer.

The expression we're tackling is:

2y+7−4y−10\frac{2y + 7}{-4y - 10}

Our goal is to find the value of this expression when $y = -3$. Let's dive into the solution step-by-step.

Step 1: Substitute 'y' with -3

The very first thing we need to do is replace every instance of 'y' in the expression with the value -3. This is a fundamental step in evaluating algebraic expressions. It's like plugging in a value into a machine to see what comes out!

So, let's do it:

2(−3)+7−4(−3)−10\frac{2(-3) + 7}{-4(-3) - 10}

Notice how we've replaced 'y' with '(-3)'. The parentheses are crucial here, especially when dealing with negative numbers. They ensure that we multiply the entire number, including its sign.

Step 2: Simplify the Numerator

The numerator is the top part of our fraction, which is currently $2(-3) + 7$. We need to simplify this using the order of operations (PEMDAS/BODMAS). Remember, multiplication comes before addition.

First, let's multiply:

2×(−3)=−62 \times (-3) = -6

Now, we have:

−6+7-6 + 7

Adding these together:

−6+7=1-6 + 7 = 1

So, the simplified numerator is 1. We're one step closer to solving this!

Step 3: Simplify the Denominator

Now, let's focus on the denominator, which is the bottom part of the fraction: $-4(-3) - 10$. Again, we need to follow the order of operations.

First, let's multiply:

−4×(−3)=12-4 \times (-3) = 12

Remember, a negative times a negative equals a positive. This is a crucial rule to keep in mind when working with negative numbers.

Now, our denominator looks like this:

12−1012 - 10

Subtracting gives us:

12−10=212 - 10 = 2

So, the simplified denominator is 2.

Step 4: Form the Simplified Fraction

Now that we've simplified both the numerator and the denominator, we can put them together to form our simplified fraction. The numerator is 1, and the denominator is 2, so our fraction is:

12\frac{1}{2}

This is the value of the expression when $y = -3$.

Step 5: Check the Answer Choices

Okay, we've got our answer: $\frac{1}{2}$. Now, let's look at the answer choices provided in the question and see if our answer matches any of them.

The answer choices were:

A. $ rac{-1}{22}$ B. $ rac{-13}{2}$ C. $ rac{1}{2}$ D. 1

Looking at the options, we can see that our answer, $\frac{1}{2}$, matches option C. So, C is the correct answer!

Conclusion

Alright, we did it! We successfully evaluated the expression $\frac{2y + 7}{-4y - 10}$ when $y = -3$. We carefully substituted the value of 'y', simplified the numerator and denominator separately, and then formed the simplified fraction. Our final answer was $\frac{1}{2}$, which corresponds to answer choice C.

Key Takeaways:

  • Substitution is a fundamental concept in algebra. It's the process of replacing a variable with its given value.
  • The order of operations (PEMDAS/BODMAS) is crucial for simplifying expressions correctly. Remember: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Pay close attention to signs, especially when dealing with negative numbers. A negative times a negative is a positive, and a negative times a positive is a negative.
  • Always double-check your work to avoid careless errors.

I hope this step-by-step guide helped you understand how to evaluate algebraic expressions. Keep practicing, and you'll become a pro in no time!