Solving 3y-7: A Step-by-Step Guide

Hey guys! Ever found yourself staring at an equation like 3y - 7 and wondering, "What am I supposed to do with this?" Don't worry, you're not alone! Math can seem intimidating, but breaking it down step by step makes it super manageable. In this guide, we're going to walk through how to solve equations like this, making sure you understand the process along the way. Let's dive in!

Understanding the Basics of Algebraic Equations

Before we jump into solving 3y - 7, let's quickly recap some fundamental concepts. Think of an algebraic equation as a balanced scale. On one side, you have an expression (like our 3y - 7), and on the other side, you have a value or another expression. The goal is to figure out what value of 'y' keeps the scale balanced. In other words, we want to isolate 'y' on one side of the equation.

• Variables: These are letters (like 'y' in our case) that represent unknown values. Our mission is to find out what these values are!
• Constants: These are the numbers in the equation (like '-7'). They don't change their value.
• Coefficients: This is the number multiplied by the variable (like '3' which is multiplied by 'y').
• Operations: These are the mathematical actions we perform (addition, subtraction, multiplication, division). The key thing to remember is that whatever operation you do on one side of the equation, you must do on the other side to maintain balance. It’s like adding or removing the same weight from both sides of a scale – it stays balanced!

Step-by-Step Solution for 3y - 7 = 0

Okay, let's get down to business and solve our equation, 3y - 7 = 0. Yes, I added the "= 0" part; this completes the equation, making it something we can actually solve. Without the "= 0" (or any other value), it's just an expression, not an equation.

Step 1: Isolate the Term with the Variable

The first thing we want to do is get the term with the variable (3y) by itself on one side of the equation. Currently, we have "- 7" hanging around on the same side. To get rid of it, we need to perform the opposite operation. Since we're subtracting 7, we'll add 7 to both sides of the equation:

3y - 7 + 7 = 0 + 7

This simplifies to:

3y = 7

Awesome! Now we have 3y isolated on the left side. We're one step closer!

Step 2: Solve for the Variable

Now we have 3y = 7. Remember, 3y means "3 multiplied by y". To isolate 'y', we need to undo this multiplication. The opposite of multiplication is division, so we'll divide both sides of the equation by 3:

3y / 3 = 7 / 3

This simplifies to:

y = 7/3

There you have it! We've solved for 'y'. The value of 'y' that makes the equation 3y - 7 = 0 true is 7/3. You can also express this as a mixed number (2 1/3) or a decimal (approximately 2.33).

Checking Your Answer

One of the best things about algebra is that you can always check your answer! To do this, simply substitute the value you found for 'y' back into the original equation and see if it holds true. Let's try it with y = 7/3:

3 * (7/3) - 7 = 0

The 3's cancel out:

7 - 7 = 0

0 = 0

It works! Since the equation holds true, we know our answer is correct. This is a great habit to get into, especially on tests, to ensure you're getting those points!

Common Mistakes to Avoid

Solving equations is a skill that gets better with practice, but here are a few common pitfalls to watch out for:

• Not Performing Operations on Both Sides: This is the biggest mistake people make! Remember, the equation is a balanced scale. If you add, subtract, multiply, or divide on one side, you must do the same on the other side.
• Incorrect Order of Operations: Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Make sure you're performing operations in the correct order.
• Sign Errors: Keep a close eye on your positive and negative signs, especially when dealing with subtraction and negative numbers.
• Forgetting to Check Your Answer: As we showed earlier, checking your answer is a fantastic way to catch mistakes and ensure accuracy.

Practice Problems

Ready to put your skills to the test? Try solving these equations on your own:

1. 5x + 2 = 17
2. 2y - 9 = -1
3. 4z + 3 = 15

Write down your solutions, check them, and you'll be a pro in no time!

Beyond Basic Equations

We've covered the basics of solving a simple linear equation. But the world of algebra is vast and exciting! As you progress, you'll encounter more complex equations involving:

• Distributive Property: Equations like 2(x + 3) = 10 require you to distribute the 2 before solving.
• Combining Like Terms: Equations like 3x + 2 + x = 9 involve combining terms with the same variable (3x and x).
• Equations with Variables on Both Sides: Equations like 4x - 5 = 2x + 1 require you to move variables to one side and constants to the other.
• Quadratic Equations: Equations like x² + 3x + 2 = 0 involve a variable raised to the power of 2 and require different solving techniques (like factoring or the quadratic formula).

Each of these builds upon the fundamental principles we've discussed here, so mastering the basics is crucial!

Conclusion: You've Got This!

Solving equations like 3y - 7 = 0 might seem daunting at first, but with a clear understanding of the fundamentals and a step-by-step approach, it becomes much more manageable. Remember to isolate the variable, perform operations on both sides of the equation, and always check your answer. With practice and persistence, you'll become a math whiz in no time! Keep practicing, and don't be afraid to ask for help when you need it. You got this! And if you get stuck again, remember this guide – it's here for you. Happy solving!