Solving 28x - 56 = 22x - 65: A Step-by-Step Guide

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Hey guys! Today, we're going to break down how to solve the equation 28x - 56 = 22x - 65. It might look a little intimidating at first, but don't worry, we'll take it step by step. We'll focus on understanding the underlying principles and applying them methodically to reach the solution. So, grab your pencils and let's get started!

Step 1: Isolating the Variable Term

The first crucial step in solving any algebraic equation is to isolate the variable term. In our case, the variable is 'x'. What does isolating the variable term really mean, you ask? Well, think of it as gathering all the 'x' terms on one side of the equation and all the constant terms (the numbers) on the other side. This simplifies the equation and brings us closer to finding the value of 'x'.

So, how do we do this? Our equation is 28x - 56 = 22x - 65. We need to get the 'x' terms together. A common strategy is to move the 'x' term from the right side of the equation to the left side. To do this, we use the principle of equality, which states that we can perform the same operation on both sides of an equation without changing its balance. In simpler terms, whatever we do to one side, we must do to the other.

In this case, we'll subtract 22x from both sides of the equation. This will eliminate the 'x' term on the right side and move it to the left. Here's how it looks:

28x - 56 - 22x = 22x - 65 - 22x

Now, we simplify. On the left side, 28x - 22x becomes 6x. On the right side, 22x - 22x cancels out, leaving us with just -65. So, our equation now looks like this:

6x - 56 = -65

Great! We've successfully moved the 'x' term to the left side. Notice how we used the subtraction property of equality to maintain the balance of the equation. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems with confidence. Remember, the goal is to simplify and get the variable term by itself. We're one step closer to solving for 'x'!

Step 2: Isolating the Variable

Now that we have 6x - 56 = -65, it's time to isolate the variable 'x' completely. Remember, our ultimate goal is to get 'x' by itself on one side of the equation. To do this, we need to get rid of the -56 that's hanging out on the left side with the 'x' term. Think of it like unwrapping a present – we need to peel away the layers to reveal what's inside, which in this case is the value of 'x'.

How do we get rid of the -56? We use the inverse operation. The inverse operation of subtraction is addition. So, we'll add 56 to both sides of the equation. Why both sides? Because, as we learned in the first step, we need to maintain the balance of the equation. Whatever we do to one side, we must do to the other. This is the golden rule of algebra!

So, let's add 56 to both sides:

6x - 56 + 56 = -65 + 56

Now, we simplify. On the left side, -56 + 56 cancels out, leaving us with just 6x. On the right side, -65 + 56 equals -9. Our equation now looks like this:

6x = -9

Awesome! We're getting closer. The 'x' term is almost completely isolated. There's just one more step: getting rid of the 6 that's multiplying the 'x'.

Step 3: Solving for x

We've reached the final stage! We have 6x = -9, and our mission is to find the value of 'x'. Remember, 6x means 6 multiplied by 'x'. So, to isolate 'x', we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we'll divide both sides of the equation by 6.

Why 6? Because dividing 6x by 6 will leave us with just 'x' on the left side. And, as always, we need to do the same thing to both sides to maintain the balance of the equation. You guys are probably tired of hearing that, but it's so important!

Let's divide both sides by 6:

6x / 6 = -9 / 6

Now, we simplify. On the left side, 6x / 6 simplifies to 'x'. On the right side, -9 / 6 simplifies to -3/2 (we can reduce the fraction by dividing both the numerator and denominator by 3). So, our equation now looks like this:

x = -3/2

Or, if you prefer decimals, -3/2 is equal to -1.5.

x = -1.5

Woohoo! We did it! We've successfully solved for 'x'. The solution to the equation 28x - 56 = 22x - 65 is x = -3/2 or x = -1.5.

Recap and Key Takeaways

Let's quickly recap the steps we took to solve the equation:

  1. Isolate the variable term: We subtracted 22x from both sides to get 6x - 56 = -65.
  2. Isolate the variable: We added 56 to both sides to get 6x = -9.
  3. Solve for x: We divided both sides by 6 to get x = -3/2 or x = -1.5.

The key takeaways from this exercise are:

  • The importance of the principle of equality: Whatever operation you perform on one side of the equation, you must perform on the other.
  • Using inverse operations to isolate the variable: Subtraction undoes addition, and division undoes multiplication.
  • Simplifying as you go: This makes the equation easier to work with and reduces the chance of errors.

Solving equations is like following a recipe. Once you understand the steps and the ingredients (the principles of algebra), you can apply them to a wide variety of problems. Keep practicing, and you'll become a pro in no time!

Remember guys, mathematics isn't about memorizing formulas; it's about understanding concepts and applying them logically. So, keep exploring, keep questioning, and keep solving! You've got this!