Solving X + 0.7 = 1 - 0.2x Two Different Ways

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Hey guys! Today, we're diving into the world of algebra to tackle a fun little equation: x + 0.7 = 1 - 0.2x. But we're not just going to solve it once; we're going to solve it in two different ways. That's right, we're flexing our math muscles and showing off our problem-solving skills. So, buckle up, grab your pencils, and let's get started!

Method 1: The Classic Approach – Isolating x

Our first method is the classic approach to solving linear equations: isolating the variable. This means we want to get all the 'x' terms on one side of the equation and all the constant terms (the numbers) on the other side. Think of it like sorting your socks – you want all the same colors together, right? We're doing the same thing with our equation.

First, let's focus on getting all the 'x' terms on the left side. We currently have 'x' on the left and '-0.2x' on the right. To get rid of the '-0.2x' on the right, we need to do the opposite operation: addition. We'll add '0.2x' to both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level.

So, we have:

x + 0.7 + 0.2x = 1 - 0.2x + 0.2x

Now, let's simplify. On the left side, we can combine the 'x' terms: 'x + 0.2x = 1.2x'. On the right side, '-0.2x + 0.2x' cancels out, leaving us with just '1'. Our equation now looks like this:

  1. 2x + 0.7 = 1

Great! We've got all our 'x' terms on the left. Now, let's move the constant terms to the right. We have '+0.7' on the left, so we'll do the opposite operation: subtraction. We'll subtract '0.7' from both sides:

  1. 2x + 0.7 - 0.7 = 1 - 0.7

Simplifying again, '0.7 - 0.7' cancels out on the left, and '1 - 0.7 = 0.3' on the right. Our equation is now:

  1. 2x = 0.3

We're almost there! We have '1.2x' on the left, and we want just 'x'. To do this, we'll divide both sides by '1.2'. Remember, multiplication and division are inverse operations, just like addition and subtraction.

  1. 2x / 1.2 = 0.3 / 1.2

Now, let's do the math. '1.2x / 1.2' simplifies to 'x'. And '0.3 / 1.2' equals '0.25'. So, our solution is:

x = 0.25

Boom! We've solved the equation using the classic method. But wait, there's more! We promised you two ways, and we always deliver. Let's move on to our second method.

Method 2: Clearing the Decimals – A Cleaner Approach

Our second method is all about making our lives easier. We're going to get rid of those pesky decimals right from the start. Decimals can sometimes make equations look a little intimidating, but we can banish them with a simple trick: multiplying the entire equation by a power of 10.

Look at our equation: 'x + 0.7 = 1 - 0.2x'. The decimals go to the tenths place (one decimal place). So, to clear them, we'll multiply every term in the equation by 10. This is crucial – you have to multiply everything to keep the equation balanced. It's like making a recipe – you can't just double one ingredient; you have to double them all!

Multiplying by 10, we get:

10(x + 0.7) = 10(1 - 0.2x)

Now, we'll distribute the 10 to each term inside the parentheses:

10 * x + 10 * 0.7 = 10 * 1 - 10 * 0.2x

This simplifies to:

10x + 7 = 10 - 2x

Look at that! No more decimals! Our equation is much cleaner and easier to work with. Now, we can proceed with isolating 'x' just like we did in Method 1. Let's get all the 'x' terms on the left side by adding '2x' to both sides:

10x + 7 + 2x = 10 - 2x + 2x

Simplifying, we get:

12x + 7 = 10

Next, let's move the constant terms to the right by subtracting '7' from both sides:

12x + 7 - 7 = 10 - 7

This gives us:

12x = 3

Finally, we'll divide both sides by '12' to isolate 'x':

12x / 12 = 3 / 12

This simplifies to:

x = 0.25

Ta-da! We got the same answer, 'x = 0.25', using our second method. Clearing the decimals made the equation look less scary, and the rest of the steps were the same as before.

Why Two Methods? The Power of Flexibility

You might be wondering, “Why bother learning two methods? Isn't one enough?” Well, knowing multiple ways to solve a problem is like having extra tools in your toolbox. Some methods might be easier or faster for certain types of equations. In this case, clearing the decimals can be a great strategy when you're dealing with equations that have lots of decimals. It simplifies the numbers and reduces the chance of making calculation errors.

More importantly, understanding different approaches helps you develop a deeper understanding of the underlying mathematical concepts. You're not just memorizing steps; you're learning how the rules of algebra work and how you can manipulate equations to find solutions. This is a powerful skill that will help you tackle more complex problems in the future.

Key Takeaways and Pro Tips

  • Isolate the variable: This is the core principle of solving linear equations. Get all the 'x' terms on one side and the constant terms on the other.
  • Do the same to both sides: Remember the seesaw! Any operation you perform on one side of the equation must be performed on the other to maintain balance.
  • Inverse operations are your friends: Use addition to undo subtraction, subtraction to undo addition, multiplication to undo division, and division to undo multiplication.
  • Clear decimals (or fractions): Multiplying by a power of 10 (for decimals) or the least common multiple of the denominators (for fractions) can simplify the equation.
  • Check your answer: Always plug your solution back into the original equation to make sure it works. This is a crucial step to avoid errors.

Practice Makes Perfect

The best way to master equation-solving is to practice! Try solving other equations using both methods. Experiment with different approaches and see which ones work best for you. The more you practice, the more confident you'll become in your algebra skills.

So there you have it, guys! We've successfully solved the equation 'x + 0.7 = 1 - 0.2x' in two different ways. We've learned about isolating variables, clearing decimals, and the importance of having multiple problem-solving strategies. Now, go forth and conquer those equations!

Remember, math is not just about finding the right answer; it's about understanding the process and developing your critical thinking skills. Keep practicing, keep exploring, and keep having fun with math!