Solving $3(y+2/5) = -1/5$: A Step-by-Step Guide

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Hey guys! Today, we're diving into a super common type of problem you'll see in algebra: solving a linear equation. Specifically, we're going to tackle the equation $3(y+\frac{2}{5})=-\frac{1}{5}$. Don't worry, it looks harder than it actually is! We'll break it down step-by-step so you can ace these problems every time. Grab your pencils and let's get started!

Understanding the Basics

Before we jump into solving, let's make sure we're all on the same page with some basic concepts. When we talk about solving an equation, we mean finding the value of the variable (in this case, y) that makes the equation true. Think of it like a puzzle where you need to find the missing piece that fits perfectly.

Equations have two sides, separated by an equals sign (=). Whatever you do to one side of the equation, you must do to the other side to keep the equation balanced. This is a fundamental principle in algebra and is crucial for solving equations correctly. For instance, if you add 5 to the left side, you need to add 5 to the right side as well.

Step-by-Step Solution

Now, let's get into the nitty-gritty of solving the equation $3(y+\frac{2}{5})=-\frac{1}{5}$.

Step 1: Distribute

The first thing we need to do is get rid of the parentheses. We do this by distributing the 3 across the terms inside the parentheses. Remember, distributing means multiplying the term outside the parentheses by each term inside the parentheses.

So, $3(y+\frac{2}{5})$ becomes $3 * y + 3 * \frac{2}{5}$, which simplifies to $3y + \frac{6}{5}$. Our equation now looks like this:

$3y + \frac{6}{5} = -\frac{1}{5}$

Step 2: Isolate the Term with y

Our goal is to get the term with y (which is $3y$) by itself on one side of the equation. To do this, we need to get rid of the $+\frac{6}{5}$ on the left side. We can do this by subtracting $\frac{6}{5}$ from both sides of the equation. Remember, whatever we do to one side, we have to do to the other!

$3y + \frac{6}{5} - \frac{6}{5} = -\frac{1}{5} - \frac{6}{5}$

This simplifies to:

$3y = -\frac{7}{5}$

Step 3: Solve for y

Now, we have $3y = -\frac{7}{5}$. To solve for y, we need to get rid of the 3 that's multiplying y. We can do this by dividing both sides of the equation by 3.

$\frac{3y}{3} = \frac{-\frac{7}{5}}{3}$

This simplifies to:

$y = -\frac{7}{5} * \frac{1}{3}$

Which further simplifies to:

$y = -\frac{7}{15}$

So, the solution to the equation is $y = -\frac{7}{15}$.

Verification

To make sure we got the correct answer, we can plug our solution back into the original equation and see if it holds true.

Original equation: $3(y+\frac{2}{5})=-\frac{1}{5}$

Substitute $y = -\frac{7}{15}$:

$3(-\frac{7}{15} + \frac{2}{5}) = -\frac{1}{5}$

First, let's simplify inside the parentheses. We need a common denominator to add the fractions, which is 15.

$\frac{2}{5} = \frac{2 * 3}{5 * 3} = \frac{6}{15}$

So, $-\frac{7}{15} + \frac{6}{15} = -\frac{1}{15}$

Now, plug that back into the equation:

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

Multiply:

$-\frac{3}{15} = -\frac{1}{5}$

Simplify:

$-\frac{1}{5} = -\frac{1}{5}$

The equation holds true! This confirms that our solution, $y = -\frac{7}{15}$, is correct.

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes that students often make. Here’s what to watch out for:

• Forgetting to Distribute: Make sure you multiply the term outside the parentheses by every term inside the parentheses.
• Incorrectly Adding/Subtracting Fractions: Always make sure you have a common denominator before adding or subtracting fractions.
• Not Keeping the Equation Balanced: Whatever you do to one side of the equation, you must do to the other side.
• Sign Errors: Be extra careful with negative signs. They can easily trip you up if you're not paying attention.

Practice Problems

Want to test your skills? Try solving these equations on your own:

1. $2(x + \frac{1}{4}) = \frac{3}{2}$
2. $5(z - \frac{2}{3}) = -\frac{5}{6}$
3. $4(w + \frac{3}{8}) = \frac{1}{2}$

Solving these practice problems will help reinforce the concepts we covered and build your confidence in solving linear equations.

Tips for Success

Here are a few extra tips to help you succeed in solving equations:

• Show Your Work: Write down every step of your solution. This will help you catch mistakes and make it easier to follow your reasoning.
• Check Your Answers: Always plug your solution back into the original equation to make sure it holds true.
• Practice Regularly: The more you practice, the better you'll become at solving equations. Try to do a few problems every day to keep your skills sharp.
• Stay Organized: Keep your work neat and organized. This will make it easier to follow your steps and avoid mistakes.
• Understand the Concepts: Don't just memorize the steps. Make sure you understand why you're doing each step.

Conclusion

So there you have it, folks! Solving the equation $3(y+\frac{2}{5})=-\frac{1}{5}$ is a breeze when you break it down step by step. Remember to distribute, isolate the variable, and always check your answers. Keep practicing, and you'll become a pro at solving linear equations in no time. Now go out there and conquer those equations! You got this!

If you found this guide helpful, give it a share and let your friends know! And don't forget to check out our other math tutorials for more tips and tricks. Happy solving!