Simplify The Expression:${ \frac{y-1}{3y+15} - \frac{y+3}{5y+25} }$

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Introduction

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In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with fractions, combining them can be a bit challenging, but with the right approach, it can be a breeze. In this article, we will focus on simplifying the expression $\frac{y-1}{3y+15} - \frac{y+3}{5y+25}$ using a step-by-step approach.

Understanding the Expression

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Before we dive into simplifying the expression, let's break it down and understand what we're dealing with. The given expression is a combination of two fractions:

$\frac{y-1}{3y+15} - \frac{y+3}{5y+25}$

We can see that both fractions have a variable $y$ in the numerator and a polynomial in the denominator. Our goal is to simplify this expression by combining the two fractions.

Step 1: Factor the Denominators

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To simplify the expression, we need to factor the denominators of both fractions. Let's start with the first fraction:

$\frac{y-1}{3y+15}$

We can factor the denominator as follows:

$3y+15 = 3(y+5)$

So, the first fraction becomes:

$\frac{y-1}{3(y+5)}$

Now, let's factor the denominator of the second fraction:

$\frac{y+3}{5y+25}$

We can factor the denominator as follows:

$5y+25 = 5(y+5)$

So, the second fraction becomes:

$\frac{y+3}{5(y+5)}$

Step 2: Find the Least Common Multiple (LCM)

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To combine the two fractions, we need to find the least common multiple (LCM) of their denominators. The LCM of $3(y+5)$ and $5(y+5)$ is $15(y+5)$.

Step 3: Rewrite the Fractions with the LCM

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Now that we have the LCM, we can rewrite both fractions with the LCM as the denominator:

$\frac{y-1}{3(y+5)} = \frac{(y-1) \cdot 5}{3(y+5) \cdot 5} = \frac{5y-5}{15(y+5)}$

$\frac{y+3}{5(y+5)} = \frac{(y+3) \cdot 3}{5(y+5) \cdot 3} = \frac{3y+9}{15(y+5)}$

Step 4: Combine the Fractions

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Now that both fractions have the same denominator, we can combine them by adding or subtracting the numerators:

$\frac{5y-5}{15(y+5)} - \frac{3y+9}{15(y+5)} = \frac{(5y-5) - (3y+9)}{15(y+5)}$

Step 5: Simplify the Numerator

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Let's simplify the numerator by combining like terms:

$(5y-5) - (3y+9) = 5y-5-3y-9 = 2y-14$

So, the expression becomes:

$\frac{2y-14}{15(y+5)}$

Conclusion

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In this article, we simplified the expression $\frac{y-1}{3y+15} - \frac{y+3}{5y+25}$ using a step-by-step approach. We factored the denominators, found the least common multiple (LCM), rewrote the fractions with the LCM, combined the fractions, and simplified the numerator. By following these steps, we were able to simplify the expression and arrive at the final answer.

Final Answer

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$\frac{2y-14}{15(y+5)}$

Tips and Tricks

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• When simplifying expressions, always start by factoring the denominators.
• Finding the least common multiple (LCM) of the denominators can help you combine fractions.
• When combining fractions, make sure to add or subtract the numerators.
• Simplifying the numerator can help you arrive at the final answer.

Related Topics

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• Simplifying expressions with variables
• Factoring polynomials
• Finding the least common multiple (LCM)
• Combining fractions

References

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• Mathway
• Khan Academy
• Wolfram Alpha
  

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Introduction

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In our previous article, we simplified the expression $\frac{y-1}{3y+15} - \frac{y+3}{5y+25}$ using a step-by-step approach. However, we know that practice makes perfect, and the best way to learn is by doing. In this article, we will provide a Q&A guide to help you practice simplifying expressions.

Q&A: Simplifying Expressions

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Q1: What is the first step in simplifying an expression?

A1: The first step in simplifying an expression is to factor the denominators.

Q2: How do I find the least common multiple (LCM) of two fractions?

A2: To find the LCM of two fractions, you need to find the smallest multiple that both fractions can divide into evenly. You can do this by listing the multiples of each fraction and finding the smallest multiple that appears in both lists.

Q3: What is the difference between adding and subtracting fractions?

A3: When adding fractions, you add the numerators and keep the same denominator. When subtracting fractions, you subtract the numerators and keep the same denominator.

Q4: How do I simplify a fraction with a variable in the numerator?

A4: To simplify a fraction with a variable in the numerator, you can factor out the greatest common factor (GCF) of the numerator and denominator.

Q5: What is the final answer to the expression $\frac{y-1}{3y+15} - \frac{y+3}{5y+25}$?

A5: The final answer to the expression $\frac{y-1}{3y+15} - \frac{y+3}{5y+25}$ is $\frac{2y-14}{15(y+5)}$.

Practice Problems

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Problem 1

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Simplify the expression $\frac{x+2}{2x+6} - \frac{x-1}{3x+9}$.

Problem 2

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Simplify the expression $\frac{y+4}{3y+12} + \frac{y-2}{4y+8}$.

Problem 3

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Simplify the expression $\frac{z-3}{2z+6} - \frac{z+2}{3z+9}$.

Solutions

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Solution 1

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To simplify the expression $\frac{x+2}{2x+6} - \frac{x-1}{3x+9}$, we need to follow the same steps as before:

1. Factor the denominators: $2x+6 = 2(x+3)$ and $3x+9 = 3(x+3)$.
2. Find the LCM: The LCM of $2(x+3)$ and $3(x+3)$ is $6(x+3)$.
3. Rewrite the fractions with the LCM: $\frac{x+2}{2(x+3)} = \frac{(x+2) \cdot 3}{2(x+3) \cdot 3} = \frac{3x+6}{6(x+3)}$ and $\frac{x-1}{3(x+3)} = \frac{(x-1) \cdot 2}{3(x+3) \cdot 2} = \frac{2x-2}{6(x+3)}$.
4. Combine the fractions: $\frac{3x+6}{6(x+3)} - \frac{2x-2}{6(x+3)} = \frac{(3x+6) - (2x-2)}{6(x+3)}$.
5. Simplify the numerator: $(3x+6) - (2x-2) = 3x+6-2x+2 = x+8$.
6. The final answer is: $\frac{x+8}{6(x+3)}$.

Solution 2

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To simplify the expression $\frac{y+4}{3y+12} + \frac{y-2}{4y+8}$, we need to follow the same steps as before:

1. Factor the denominators: $3y+12 = 3(y+4)$ and $4y+8 = 4(y+2)$.
2. Find the LCM: The LCM of $3(y+4)$ and $4(y+2)$ is $12(y+4)$.
3. Rewrite the fractions with the LCM: $\frac{y+4}{3(y+4)} = \frac{(y+4) \cdot 4}{3(y+4) \cdot 4} = \frac{4y+16}{12(y+4)}$ and $\frac{y-2}{4(y+2)} = \frac{(y-2) \cdot 3}{4(y+2) \cdot 3} = \frac{3y-6}{12(y+4)}$.
4. Combine the fractions: $\frac{4y+16}{12(y+4)} + \frac{3y-6}{12(y+4)} = \frac{(4y+16) + (3y-6)}{12(y+4)}$.
5. Simplify the numerator: $(4y+16) + (3y-6) = 4y+16+3y-6 = 7y+10$.
6. The final answer is: $\frac{7y+10}{12(y+4)}$.

Solution 3

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To simplify the expression $\frac{z-3}{2z+6} - \frac{z+2}{3z+9}$, we need to follow the same steps as before:

1. Factor the denominators: $2z+6 = 2(z+3)$ and $3z+9 = 3(z+3)$.
2. Find the LCM: The LCM of $2(z+3)$ and $3(z+3)$ is $6(z+3)$.
3. Rewrite the fractions with the LCM: $\frac{z-3}{2(z+3)} = \frac{(z-3) \cdot 3}{2(z+3) \cdot 3} = \frac{3z-9}{6(z+3)}$ and $\frac{z+2}{3(z+3)} = \frac{(z+2) \cdot 2}{3(z+3) \cdot 2} = \frac{2z+4}{6(z+3)}$.
4. Combine the fractions: $\frac{3z-9}{6(z+3)} - \frac{2z+4}{6(z+3)} = \frac{(3z-9) - (2z+4)}{6(z+3)}$.
5. Simplify the numerator: $(3z-9) - (2z+4) = 3z-9-2z-4 = z-13$.
6. The final answer is: $\frac{z-13}{6(z+3)}$.

Conclusion

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In this article, we provided a Q&A guide to help you practice simplifying expressions. We also provided solutions to three practice problems to help you understand the steps involved in simplifying expressions. By following these steps and practicing regularly, you will become more confident and proficient in simplifying expressions.