Unraveling Equations: Finding 'b' In A Math Puzzle

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Unraveling Equations: Finding 'b' in a Math Puzzle

Hey math enthusiasts! Today, we're diving into a fun problem involving equations, variables, and a little bit of algebraic manipulation. We'll start with the given equations, solve for x and a, and then use those values to crack the code and find the value of b. Sounds like a plan, right? Let's get started!

Solving for x and Unveiling the First Equation: 3x - 2 = 2(4 - x)

Alright guys, let's start with the first equation: 3x - 2 = 2(4 - x). Our main goal here is to isolate x and find its value. To do this, we'll need to use our algebra skills. First, let's distribute the 2 on the right side of the equation. That gives us:

  • 3x - 2 = 8 - 2x*

Next, we'll want to get all the x terms on one side and the constants (numbers without variables) on the other. We can add 2x to both sides:

  • 3x + 2x - 2 = 8*
  • 5x - 2 = 8*

Then, add 2 to both sides:

  • 5x = 10*

Finally, divide both sides by 5 to solve for x:

  • x = 2*

So, we've successfully found the value of x! x equals 2. Not too shabby, eh?

Detailed Breakdown for Solving x

Let's break down each step of solving for x in the equation 3x - 2 = 2(4 - x):

  1. Distribution: The first step involves distributing the 2 across the terms inside the parentheses. This means multiplying 2 by both 4 and -x:
    • 2 * 4 = 8
    • 2 * -x = -2x
    • This results in the equation: 3x - 2 = 8 - 2x
  2. Combining x Terms: The next step is to get all the terms containing x on one side of the equation. To do this, add 2x to both sides. This cancels out the -2x on the right side:
    • 3x + 2x - 2 = 8 - 2x + 2x
    • This simplifies to: 5x - 2 = 8
  3. Isolating the x Term: Now, isolate the term with x. Add 2 to both sides of the equation:
    • 5x - 2 + 2 = 8 + 2
    • This results in: 5x = 10
  4. Solving for x: Finally, to find the value of x, divide both sides of the equation by 5:
    • 5x / 5 = 10 / 5
    • Therefore, x = 2

This methodical approach ensures that you accurately solve for x. Remember to always perform the same operation on both sides of the equation to maintain balance.

Determining the Value of a: Tackling the Second Equation, 2a = 3

Now, let's move on to the second equation: 2a = 3. This one is a little more straightforward. Our goal is to isolate a. To do this, we simply divide both sides of the equation by 2:

  • 2a / 2 = 3 / 2*
  • a = 3/2 or 1.5*

So, the value of a is 3/2 or 1.5. Easy peasy!

Steps to Find the Value of a

Here’s a quick rundown of how we found a:

  1. The Equation: We started with the equation 2a = 3.
  2. Isolating a: To get a by itself, we divided both sides of the equation by 2.
  3. Solution: This gave us a = 3/2, which is the same as 1.5.

See? A piece of cake! Now we have the values for both x and a.

Unveiling the Equation: 3a + 1

At this point, this part isn't really necessary, as the problem is clearly asking us to solve for the equation 2a = 3. It can be easily solved to find the value of a.

Putting It All Together: Solving for b in bx - a = a + x

Okay, now comes the fun part! We've found that x = 2 and a = 3/2. Now we need to use these values in the equation: bx - a = a + x to find the value of b. Let's substitute the values of x and a into the equation:

  • b(2) - (3/2) = (3/2) + 2*

Simplify the equation:

  • 2b - 3/2 = 3/2 + 2*

Combine the constants on the right side:

  • 2b - 3/2 = 7/2*

Add 3/2 to both sides:

  • 2b = 7/2 + 3/2*
  • 2b = 10/2*
  • 2b = 5*

Finally, divide both sides by 2 to solve for b:

  • b = 5/2 or 2.5*

And there you have it! The value of b is 5/2 or 2.5. We did it, guys!

Detailed Steps to Solve for b

Here's a step-by-step breakdown:

  1. Substitution: We start with the equation bx - a = a + x. We know that x = 2 and a = 3/2. Substitute these values into the equation:
    • b(2) - 3/2 = 3/2 + 2
  2. Simplify: Simplify the equation:
    • 2b - 3/2 = 7/2
  3. Isolate b: Add 3/2 to both sides:
    • 2b = 7/2 + 3/2
    • 2b = 10/2
    • 2b = 5
  4. Solve for b: Divide both sides by 2:
    • b = 5/2 or b = 2.5

Following these steps ensures accuracy when solving for b.

Summary and Key Takeaways

Alright, let's recap what we've done and highlight some key takeaways from this problem:

  • Found x: We solved the equation 3x - 2 = 2(4 - x) and found that x = 2.
  • Found a: We solved the equation 2a = 3 and found that a = 3/2.
  • Found b: We substituted the values of x and a into the equation bx - a = a + x and determined that b = 5/2.

Key Concepts and Skills Applied:

  • Solving Linear Equations: We practiced solving linear equations with one variable, which is a fundamental skill in algebra.
  • Distribution: We applied the distributive property to simplify expressions.
  • Substitution: We substituted known values into an equation to solve for an unknown variable.
  • Algebraic Manipulation: We utilized basic algebraic manipulations to isolate variables and solve for their values.

This problem reinforces the importance of understanding basic algebraic principles and how to apply them to solve for unknown variables. Keep practicing, and you'll become a pro in no time!

Final Answer

So, the answer is b = 5/2 or 2.5. We successfully navigated the equations, found the values of x, a, and finally, b. Great job, everyone! Keep up the awesome work, and keep exploring the wonderful world of mathematics. Until next time, happy calculating!