Solving Equations: Step-by-Step Guide & Solution Verification

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Hey guys! Today, we're going to dive deep into solving a specific type of equation and, more importantly, verifying our solution to make sure we've got it right. This is a crucial skill in mathematics, and mastering it will definitely boost your confidence. We'll be tackling the equation: (6x + 7)/3 = (3x + 8)/2. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into the solution, let's break down what we're actually looking at. This equation involves fractions, which might seem intimidating at first, but don't worry! We'll handle them using some simple algebraic techniques. Our goal is to isolate 'x' on one side of the equation. This means we want to manipulate the equation until we have something like x = [some number]. That "some number" will be our solution.

Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced. This is a fundamental principle in algebra. We can add, subtract, multiply, or divide both sides by the same value (except zero, of course) without changing the equation's truth.

The equation (6x + 7)/3 = (3x + 8)/2 represents a balance between two expressions. The left side, (6x + 7) divided by 3, must equal the right side, (3x + 8) divided by 2. Our task is to find the value of 'x' that makes this balance true. Think of it like a scale – we need to find the 'x' that makes both sides weigh the same.

This type of equation is often called a linear equation because, when graphed, it represents a straight line. Solving linear equations is a cornerstone of algebra, and the techniques we'll use here apply to many other problems. Understanding how to manipulate these equations and isolate the variable is a powerful tool in your mathematical arsenal. So, let's move on to the first step in solving this particular equation – clearing those fractions!

Step 1: Clearing the Fractions

Fractions can sometimes make equations look more complicated than they really are. The good news is, we can easily get rid of them! The key is to find the least common multiple (LCM) of the denominators. In our equation, the denominators are 3 and 2. The LCM of 3 and 2 is 6. This is the smallest number that both 3 and 2 divide into evenly.

To clear the fractions, we'll multiply both sides of the equation by the LCM, which is 6. This might seem like a big step, but it's actually quite straightforward:

  • 6 * [(6x + 7)/3] = 6 * [(3x + 8)/2]

Now, let's simplify each side. On the left side, 6 divided by 3 is 2. So we have:

  • 2 * (6x + 7)

On the right side, 6 divided by 2 is 3. So we have:

  • 3 * (3x + 8)

Our equation now looks much cleaner:

  • 2(6x + 7) = 3(3x + 8)

See? No more fractions! We've successfully transformed the equation into a more manageable form. By multiplying both sides by the LCM, we've eliminated the denominators and paved the way for the next step: distributing the multiplication. This technique is super useful for handling equations with fractions, so make sure you understand the logic behind it. Now, let's move on to distributing and simplifying further.

Step 2: Distributing and Simplifying

Now that we've cleared the fractions, it's time to distribute the numbers outside the parentheses. Remember the distributive property? It states that a(b + c) = ab + ac. We're going to apply this property to both sides of our equation.

On the left side, we have 2(6x + 7). Distributing the 2 gives us:

  • 2 * 6x + 2 * 7 = 12x + 14

On the right side, we have 3(3x + 8). Distributing the 3 gives us:

  • 3 * 3x + 3 * 8 = 9x + 24

Our equation now looks like this:

  • 12x + 14 = 9x + 24

We've successfully distributed and simplified both sides of the equation. The next step is to gather the 'x' terms on one side and the constant terms (the numbers without 'x') on the other side. This will help us isolate 'x' and eventually find its value. Gathering like terms is a crucial step in solving equations, as it helps to organize and simplify the equation further. So, let's move on to the next step – isolating those 'x' terms!

Step 3: Isolating the 'x' Terms

Our goal now is to get all the terms with 'x' on one side of the equation and all the constant terms on the other side. Let's choose to move the 'x' terms to the left side. To do this, we need to subtract 9x from both sides of the equation:

  • 12x + 14 - 9x = 9x + 24 - 9x

This simplifies to:

  • 3x + 14 = 24

Now, let's move the constant terms to the right side. We'll subtract 14 from both sides:

  • 3x + 14 - 14 = 24 - 14

This simplifies to:

  • 3x = 10

We're getting closer! We've successfully isolated the 'x' term on the left side. Now, only one step remains: dividing to solve for 'x'. This process of isolating the variable is at the heart of solving equations. By strategically adding, subtracting, multiplying, or dividing, we can manipulate the equation to reveal the value of 'x'. So, let's finish the job and find our solution!

Step 4: Solving for 'x'

We're almost there! We have 3x = 10. To solve for 'x', we need to divide both sides of the equation by 3:

  • (3x) / 3 = 10 / 3

This simplifies to:

  • x = 10/3

Congratulations! We've found our solution. x is equal to 10/3. This is a fraction, which is perfectly acceptable. It means that if we substitute 10/3 for 'x' in the original equation, both sides should be equal. But before we celebrate, we need to do one crucial thing: verify our solution. This is the best way to ensure we haven't made any mistakes along the way.

Finding the solution is a great accomplishment, but verifying it is just as important. It's like double-checking your work on a test – it gives you peace of mind and helps you catch any errors. So, let's move on to the final step: verifying our solution.

Step 5: Verifying the Solution

This is the most important step to ensure we didn't make any mistakes! To verify our solution, we'll substitute x = 10/3 back into the original equation:

  • (6(10/3) + 7) / 3 = (3(10/3) + 8) / 2

Let's simplify the left side first:

  • 6 * (10/3) = 20
  • 20 + 7 = 27
  • 27 / 3 = 9

So, the left side of the equation equals 9.

Now, let's simplify the right side:

  • 3 * (10/3) = 10
  • 10 + 8 = 18
  • 18 / 2 = 9

The right side of the equation also equals 9!

Since both sides are equal, our solution, x = 10/3, is correct. We've successfully solved the equation and verified our answer. This process of substitution and simplification is crucial in verifying solutions, ensuring accuracy, and building confidence in your mathematical abilities. So, always remember to check your work!

Conclusion

We did it! We successfully solved the equation (6x + 7)/3 = (3x + 8)/2 and verified our solution. We found that x = 10/3. Remember the key steps:

  1. Clear the fractions by multiplying both sides by the LCM of the denominators.
  2. Distribute and simplify both sides of the equation.
  3. Isolate the 'x' terms on one side and the constant terms on the other.
  4. Solve for 'x' by dividing both sides by the coefficient of 'x'.
  5. Verify your solution by substituting it back into the original equation.

By following these steps, you can confidently solve similar equations. Keep practicing, and you'll become a pro at solving equations! Remember, math is like any other skill – the more you practice, the better you get. So, keep challenging yourself, keep learning, and most importantly, have fun with it! You've got this!