Solving For B: (b+4)/5 = 7/4 - A Step-by-Step Guide

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Hey guys! Ever stumbled upon an equation that looks a bit intimidating but is actually quite simple once you break it down? Today, we're going to tackle one such equation: (b+4)/5 = 7/4. Don't worry, it's not as scary as it looks! We'll go through each step together, making sure you understand the logic behind it. By the end of this guide, you'll be able to solve similar equations with confidence. So, let's dive in and make some math magic happen!

Understanding the Equation

Before we jump into solving, let's understand what the equation (b+4)/5 = 7/4 actually means. In essence, it's a statement that two fractions are equal. Our mission is to find the value of 'b' that makes this statement true.

  • The left side of the equation, (b+4)/5, represents a fraction where the numerator is (b+4) and the denominator is 5. Think of 'b' as a mystery number we need to uncover.
  • The right side, 7/4, is a simple fraction.
  • The equals sign (=) is the key here! It tells us that whatever value we get on the left side must be the same as the value on the right side. This equality is what allows us to manipulate the equation and isolate 'b'.

To truly grasp this, let’s visualize it. Imagine you have a scale perfectly balanced. On one side, you have a weight represented by (b+4)/5, and on the other side, you have a weight represented by 7/4. Our goal is to figure out the exact value of 'b' that keeps the scale balanced. Each step we take in solving the equation is like adjusting the weights on the scale, ensuring it remains balanced until we isolate 'b'. This intuitive understanding will help you remember the steps better and apply them to different equations. So, let's keep this balanced scale analogy in mind as we move forward and start unwrapping this mathematical puzzle!

Step 1: Cross-Multiplication - The Great Equalizer

The first step in solving (b+4)/5 = 7/4 is to get rid of those pesky fractions. We do this using a technique called cross-multiplication. This method is a fantastic shortcut that works when you have an equation with a fraction on each side. Think of it as a neat trick to untangle the equation and make it easier to work with.

So, how does cross-multiplication actually work? It's quite simple: you multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. In our case:

  • We multiply (b+4) by 4.
  • We multiply 5 by 7.

This gives us a new equation: 4(b+4) = 5 * 7. Notice how we've eliminated the fractions! This equation is much easier to handle. But why does this work? The magic of cross-multiplication lies in the fundamental properties of equality. When you multiply both sides of an equation by the same value, the equation remains balanced. Cross-multiplication is essentially a shortcut for multiplying both sides by both denominators. It ensures that we maintain the equality while simplifying the equation. This step is crucial because it sets the stage for isolating 'b'. We've transformed the original fractional equation into a simpler, linear equation. Now, let's move on to the next step and continue our quest to solve for 'b'. Remember, each step brings us closer to unveiling the mystery number!

Step 2: Distribute the Magic - Expanding the Equation

Now that we have 4(b+4) = 5 * 7, the next step involves simplifying the left side of the equation. We need to get rid of those parentheses! This is where the distributive property comes to our rescue. Think of it as distributing happiness – or in this case, the number 4 – to everyone inside the parentheses. The distributive property states that a(b + c) = ab + ac. In simpler terms, it means we multiply the number outside the parentheses by each term inside the parentheses.

Applying this to our equation, we multiply 4 by b and 4 by 4:

  • 4 * b = 4b
  • 4 * 4 = 16

So, 4(b+4) becomes 4b + 16. Now, let's simplify the right side of the equation as well:

  • 5 * 7 = 35

Our equation now looks like this: 4b + 16 = 35. See how much cleaner it looks? We've successfully expanded the equation and eliminated the parentheses. This step is essential because it separates the 'b' term, making it easier to isolate. By distributing, we're essentially unwrapping the equation layer by layer, bringing us closer to the solution. Think of it like peeling an onion – each layer we remove reveals more of the core. We're doing the same with our equation, simplifying it until we can finally reveal the value of 'b'. So, let's keep peeling and move on to the next step, where we'll further isolate 'b'.

Step 3: Isolating 'b' - The Art of Rearranging

We're making great progress! Our equation is now 4b + 16 = 35. The goal now is to isolate 'b' on one side of the equation. Think of it as giving 'b' some personal space. To do this, we need to get rid of the +16 that's hanging out with the 4b term. The key to isolating variables is to perform the inverse operation. What's the opposite of addition? Subtraction!

So, we need to subtract 16 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. It's like a seesaw – if you take weight off one side, you need to take the same weight off the other to keep it level.

Subtracting 16 from both sides gives us:

  • 4b + 16 - 16 = 35 - 16

Simplifying this, we get:

  • 4b = 19

Look at that! The +16 is gone from the left side, and we're one step closer to getting 'b' all by itself. This step is crucial because it narrows down the possibilities for 'b'. We've effectively reduced the equation to a much simpler form, making it easier to pinpoint the exact value of our mystery variable. Now, 'b' is almost completely isolated. We just have one more step to go! Let's move on and finish the job, giving 'b' the solo spotlight it deserves.

Step 4: The Grand Finale - Dividing to Conquer

We're in the home stretch! Our equation is currently 4b = 19. 'b' is almost completely isolated, but it's still attached to that 4. Remember, 4b means 4 multiplied by b. So, to get 'b' all by itself, we need to do the inverse operation of multiplication, which is division. We need to divide both sides of the equation by 4.

Dividing both sides by 4 gives us:

  • 4b / 4 = 19 / 4

Simplifying this, we get:

  • b = 19/4

And there you have it! We've successfully solved for 'b'! The value of 'b' that makes the equation (b+4)/5 = 7/4 true is 19/4. This can also be expressed as a mixed number (4 3/4) or a decimal (4.75). This step is the culmination of all our efforts. We've meticulously unwrapped the equation, step by step, until we've revealed the precise value of 'b'. Think of it as reaching the center of a maze – after navigating through all the twists and turns, we've finally arrived at our destination. Congratulations! You've conquered the equation and found the solution. But our journey doesn't end here. It's always a good idea to check our work to ensure we've solved it correctly. So, let's move on to the final, crucial step: verifying our answer.

Step 5: Double-Checking Our Victory - Verification Time

We've found that b = 19/4, but before we celebrate our victory, it's always wise to double-check our work. Think of this as the final quality assurance step – we want to make sure our answer is rock solid! The best way to verify our solution is to substitute the value we found for 'b' back into the original equation: (b+4)/5 = 7/4.

Let's plug in b = 19/4:

  • ((19/4) + 4) / 5 = 7/4

Now, we need to simplify the left side of the equation. First, let's deal with the addition inside the parentheses. To add 19/4 and 4, we need to express 4 as a fraction with a denominator of 4. 4 is the same as 16/4. So:

  • (19/4) + (16/4) = 35/4

Now, our equation looks like this:

  • (35/4) / 5 = 7/4

Dividing by 5 is the same as multiplying by 1/5. So:

  • (35/4) * (1/5) = 35/20

Now, let's simplify the fraction 35/20. Both 35 and 20 are divisible by 5:

  • 35/20 = 7/4

So, our equation now looks like this:

  • 7/4 = 7/4

It checks out! The left side of the equation equals the right side. This confirms that our solution, b = 19/4, is correct. This verification step is incredibly important. It gives us peace of mind knowing that we haven't made any mistakes along the way. Think of it as the final piece of the puzzle clicking into place, completing the picture and confirming our success. You've not only solved the equation, but you've also proven that your solution is accurate. That's something to be proud of! You've truly mastered this equation, from start to finish.

Conclusion: You've Cracked the Code!

Woohoo! You've successfully solved the equation (b+4)/5 = 7/4, and you've even verified your answer. Give yourself a pat on the back – you've earned it! We've walked through each step together, from understanding the equation to the grand finale of verification. You've learned how to:

  • Cross-multiply to eliminate fractions.
  • Distribute to expand equations.
  • Isolate variables by performing inverse operations.
  • Verify your solution to ensure accuracy.

These are powerful skills that you can apply to a wide range of algebraic equations. Remember, the key to mastering math is practice. The more you solve equations, the more comfortable and confident you'll become. So, don't be afraid to tackle new challenges and explore different types of problems. Math is like a puzzle – each equation is a new challenge waiting to be solved. And with the tools and techniques you've learned today, you're well-equipped to crack the code! Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!