Solving For Y: A Step-by-Step Guide To The Equation

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Hey guys! Today, we're diving into a common algebraic problem: solving for a variable, specifically 'y'. We'll break down the equation (7y + 6) / 2 = (3y - 5) / 5 + 33 step-by-step. So, if you've ever felt a little lost with these types of problems, stick around – we're going to make it crystal clear!

Understanding the Equation

Before we jump into the solution, let's understand what this equation is telling us. We have a variable, 'y', which represents an unknown value. Our goal is to isolate 'y' on one side of the equation to figure out its value. The equation involves fractions, addition, and multiplication, so we'll need to use several algebraic techniques to solve it.

The equation we're tackling is:

(7y + 6) / 2 = (3y - 5) / 5 + 33

This looks a bit intimidating at first glance, but don't worry! We'll break it down into manageable steps. Remember, the key to solving any algebraic equation is to perform the same operations on both sides, keeping the equation balanced.

So, let's get started!

Step 1: Eliminate the Fractions

Fractions can often make equations look more complicated than they are. Our first step is to get rid of them. To do this, we'll find the least common multiple (LCM) of the denominators, which are 2 and 5 in this case. The LCM of 2 and 5 is 10. We'll multiply both sides of the equation by 10:

10 * [(7y + 6) / 2] = 10 * [(3y - 5) / 5 + 33]

Now, we distribute the 10 on both sides:

10 * (7y + 6) / 2 = 10 * (3y - 5) / 5 + 10 * 33

This simplifies to:

5 * (7y + 6) = 2 * (3y - 5) + 330

See? Much cleaner already! Multiplying by the LCM effectively cancels out the denominators, leaving us with a more manageable equation.

Step 2: Distribute and Simplify

Next, we'll distribute the numbers outside the parentheses on both sides of the equation. This means multiplying the number outside the parentheses by each term inside:

5 * (7y + 6) becomes 5 * 7y + 5 * 6 = 35y + 30

2 * (3y - 5) becomes 2 * 3y - 2 * 5 = 6y - 10

So, our equation now looks like this:

35y + 30 = 6y - 10 + 330

Now, let's simplify the right side by combining the constant terms:

35y + 30 = 6y + 320

We've made good progress! We've eliminated the fractions and distributed, simplifying the equation considerably.

Step 3: Isolate the Variable Term

Our next goal is to get all the 'y' terms on one side of the equation and the constant terms on the other. To do this, we'll subtract 6y from both sides:

35y + 30 - 6y = 6y + 320 - 6y

This simplifies to:

29y + 30 = 320

Now, we have the 'y' term isolated on the left side, but we still have that pesky +30. Let's get rid of it by subtracting 30 from both sides:

29y + 30 - 30 = 320 - 30

This gives us:

29y = 290

We're getting so close! The 'y' term is almost completely isolated.

Step 4: Solve for y

Finally, to solve for 'y', we need to get 'y' by itself. Since 'y' is being multiplied by 29, we'll divide both sides of the equation by 29:

29y / 29 = 290 / 29

This simplifies to:

y = 10

And there you have it! We've solved for 'y'. The value of 'y' that satisfies the equation is 10.

Step 5: Verification (Always a Good Idea!)

To be absolutely sure we have the correct answer, we should plug y = 10 back into the original equation and see if it holds true. This is a crucial step in problem-solving – it ensures that our solution is accurate.

Original equation:

(7y + 6) / 2 = (3y - 5) / 5 + 33

Substitute y = 10:

(7 * 10 + 6) / 2 = (3 * 10 - 5) / 5 + 33

Simplify:

(70 + 6) / 2 = (30 - 5) / 5 + 33

76 / 2 = 25 / 5 + 33

38 = 5 + 33

38 = 38

It checks out! Both sides of the equation are equal, so our solution, y = 10, is correct.

Key Concepts Used

Let's quickly recap the key algebraic concepts we used to solve this equation:

  • Least Common Multiple (LCM): Finding the LCM of the denominators allowed us to eliminate fractions.
  • Distributive Property: We used this to multiply a number by an expression in parentheses.
  • Combining Like Terms: Simplifying the equation by combining constant terms and variable terms.
  • Inverse Operations: Using opposite operations (addition/subtraction, multiplication/division) to isolate the variable.
  • Substitution: Plugging the solution back into the original equation to verify its correctness.

Tips for Solving Equations

Here are a few extra tips that can help you tackle similar equations:

  • Stay Organized: Write down each step clearly. This helps you avoid mistakes and makes it easier to review your work.
  • Take it One Step at a Time: Don't try to do too much in one step. Breaking the problem down makes it less overwhelming.
  • Check Your Work: Always double-check your calculations and verify your solution.
  • Practice Makes Perfect: The more you practice, the better you'll become at solving equations. There are tons of online resources and practice problems available!

Common Mistakes to Avoid

Here are a few common mistakes people make when solving equations, so you can be sure to avoid them:

  • Forgetting to Distribute: Make sure you multiply the number outside the parentheses by every term inside.
  • Incorrectly Combining Like Terms: Pay attention to the signs (+ and -) when combining terms.
  • Not Performing Operations on Both Sides: Remember, whatever you do to one side of the equation, you must do to the other to keep it balanced.
  • Skipping Verification: It’s tempting to skip the verification step, but it’s crucial to catch any errors.

Wrapping Up

So there you have it! We've successfully solved the equation (7y + 6) / 2 = (3y - 5) / 5 + 33. Remember, solving equations is like following a recipe – if you follow the steps carefully, you'll get the right result. Keep practicing, and you'll become a master equation solver in no time!

If you have any questions or want to try another example, let me know in the comments! Happy solving!