Solving For X In The Equation -px + R = -8x - 2

Are you ready to dive into the world of algebra, guys? Today, we're going to tackle an equation and solve for the elusive variable x. It might seem daunting at first, but trust me, with a step-by-step approach, it's totally manageable. Our mission, should we choose to accept it, is to find the value of x in the equation -px + r = -8x - 2. Let's break it down and make math our friend!

Understanding the Equation: -px + r = -8x - 2

Before we jump into solving, let's get comfy with the equation itself. We have -px + r = -8x - 2. Notice that x appears on both sides of the equals sign. Our goal is to isolate x on one side so we can figure out its value. Think of it like a puzzle – we need to rearrange the pieces (terms) until we reveal the solution.

Key Components of the Equation

• -px: This term means “negative p times x.” Here, p is a coefficient, which is just a number that multiplies the variable x. We don't know the exact value of p yet, but we'll work with it algebraically.
• r: This is a constant, meaning it's a number that doesn't change. It's just a fixed value in our equation.
• -8x: Similar to -px, this is -8 times x. The coefficient here is -8.
• -2: Another constant term, just like r.

The Strategy: Isolating x

To solve for x, we'll use a few key algebraic principles:

1. Combining Like Terms: We'll group all the x terms on one side of the equation and all the constant terms on the other side.
2. Inverse Operations: We'll use addition and subtraction to move terms around, and multiplication and division to isolate x.

Step-by-Step Solution

Let's get our hands dirty and solve this equation step by step. It's like following a recipe, but instead of cookies, we're baking up the value of x!

Step 1: Gather the x Terms

Our first mission is to get all the terms with x on one side of the equation. Let's choose the left side. To do this, we'll add 8x 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.

-px + r + 8x = -8x - 2 + 8x

Simplifying this, we get:

-px + 8x + r = -2

Step 2: Gather the Constant Terms

Now, let's move the constant term r to the right side of the equation. To do this, we'll subtract r from both sides:

-px + 8x + r - r = -2 - r

Simplifying, we have:

-px + 8x = -2 - r

Step 3: Factor out x

Notice that x is a common factor on the left side of the equation. Let's factor it out. This is like reverse-distributing. We're pulling the x out and putting the remaining terms in parentheses.

x(-p + 8) = -2 - r

Step 4: Isolate x

We're almost there! To isolate x, we need to get rid of the term (-p + 8) that's multiplying it. We'll do this by dividing both sides of the equation by (-p + 8). Remember, division is the inverse operation of multiplication.

x = (-2 - r) / (-p + 8)

Step 5: Simplify (Optional)

We can simplify the expression a bit by factoring out a -1 from both the numerator and the denominator. This can make the equation look cleaner and sometimes easier to work with.

x = -(2 + r) / -(p - 8)

Since a negative divided by a negative is a positive, we can rewrite this as:

x = (2 + r) / (8 - p)

The Solution: x = (2 + r) / (8 - p)

And there you have it, folks! We've successfully solved for x. Our final answer is:

x = (2 + r) / (8 - p)

This means that the value of x depends on the values of r and p. If we knew the specific values of r and p, we could plug them into this equation and get a numerical value for x. Isn't algebra amazing?

Checking Our Work

It's always a good idea to check our solution to make sure we didn't make any mistakes along the way. To do this, we can plug our solution for x back into the original equation and see if both sides are equal. This is like double-checking your bank statement to make sure the numbers add up.

Let's substitute x = (2 + r) / (8 - p) into our original equation: -px + r = -8x - 2

-p * ((2 + r) / (8 - p)) + r = -8 * ((2 + r) / (8 - p)) - 2

This looks a bit messy, but we can simplify it. This step is crucial to ensure we didn't make any errors in our calculations. By plugging the solution back into the original equation, we're verifying its correctness.

After simplifying both sides, if they are equal, we can confidently say that our solution for x is correct. This process reinforces the accuracy of our algebraic manipulations.

Common Mistakes to Avoid

Solving for x can be tricky, and it's easy to make mistakes. Here are a few common pitfalls to watch out for:

• Forgetting to Distribute: When multiplying a term by an expression in parentheses, make sure to distribute it to every term inside the parentheses.
• Incorrectly Combining Like Terms: Only terms with the same variable and exponent can be combined.
• Dividing by Zero: Remember, division by zero is undefined. Make sure the denominator in your solution is not zero. In our case, p cannot be 8.
• Sign Errors: Pay close attention to negative signs. They can easily trip you up if you're not careful.

Real-World Applications

You might be wondering, “When will I ever use this in real life?” Well, solving for x is a fundamental skill in many areas, including:

• Physics: Calculating motion, forces, and energy.
• Engineering: Designing structures, circuits, and machines.
• Economics: Modeling supply and demand, and financial analysis.
• Computer Science: Developing algorithms and solving computational problems.

So, even though it might seem abstract now, mastering algebra can open doors to many exciting fields.

Practice Makes Perfect

The best way to become comfortable with solving for x is to practice, practice, practice! Try solving different equations with varying levels of complexity. The more you practice, the more confident you'll become. It's like learning a new language – the more you use it, the better you get.

Conclusion: You've Got This!

Solving for x in the equation -px + r = -8x - 2 might have seemed challenging at first, but we've broken it down into manageable steps. Remember, the key is to isolate x by using inverse operations and keeping the equation balanced. With a little practice and perseverance, you'll be solving equations like a pro in no time!

Keep up the great work, guys! Math is an adventure, and you're all intrepid explorers. Now go out there and conquer those equations!