Solving -17x + 13 + 6x - 23 = 10.8x: A Step-by-Step Guide

Understanding Linear Equations

Hey guys! Let's dive into the world of linear equations and tackle the problem -17x + 13 + 6x - 23 = 10.8x. Before we jump into solving this specific equation, let's quickly recap what linear equations are all about. A linear equation is basically a mathematical statement that shows the equality between two expressions. These expressions involve variables (usually represented by letters like x, y, or z) raised to the power of one. No squares, cubes, or anything fancy like that! The goal is to find the value of the variable that makes the equation true. Think of it like a puzzle where you need to figure out what number can replace the x to balance both sides of the equation.

Why are linear equations so important? Well, they pop up everywhere in real life! From calculating the cost of groceries to figuring out travel times, linear equations are the unsung heroes behind many everyday calculations. In science and engineering, they are fundamental tools for modeling relationships between different quantities. In economics, they help us understand supply and demand. So, mastering linear equations is not just about acing your math class; it’s about developing a crucial skill that will serve you well in many aspects of life.

Now, when we talk about solving a linear equation, we're essentially trying to isolate the variable on one side of the equation. We want to get something like x = a number. To do this, we use a set of algebraic rules that allow us to manipulate the equation without changing its fundamental truth. These rules include adding or subtracting the same value from both sides, multiplying or dividing both sides by the same non-zero value, and simplifying expressions by combining like terms. The key is to maintain balance – whatever you do to one side of the equation, you must do to the other. Think of it like a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle of balance is the cornerstone of solving linear equations.

Step-by-Step Solution: -17x + 13 + 6x - 23 = 10.8x

Okay, let's get our hands dirty and solve this equation: -17x + 13 + 6x - 23 = 10.8x. We'll break it down step-by-step so it's super clear.

Step 1: Combine Like Terms

First up, we need to simplify each side of the equation by combining like terms. Like terms are those that have the same variable raised to the same power (in this case, just x) or are constants (just numbers). On the left side, we have -17x and 6x, which are like terms. We also have 13 and -23, which are constants and thus also like terms. Let's combine them:

• -17x + 6x = -11x
• 13 - 23 = -10

So, the left side of the equation simplifies to -11x - 10. Our equation now looks like this: -11x - 10 = 10.8x

Step 2: Move Variables to One Side

Next, we want to get all the terms with x on one side of the equation. It doesn't matter which side we choose, but let's move them to the right side in this case. To do this, we'll add 11x to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain balance:

• -11x - 10 + 11x = 10.8x + 11x
• -10 = 21.8x

Now, our equation is -10 = 21.8x. We're getting closer!

Step 3: Isolate the Variable

Our final goal is to isolate x, meaning we want to get x all by itself on one side of the equation. Right now, x is being multiplied by 21.8. To undo this multiplication, we'll divide both sides of the equation by 21.8:

• -10 / 21.8 = 21.8x / 21.8
• x = -10 / 21.8

Step 4: Simplify the Solution

We have our solution, but it's in fraction form. Let's simplify it. We can divide -10 by 21.8 using a calculator to get a decimal approximation:

• x ≈ -0.4587

So, the solution to the equation -17x + 13 + 6x - 23 = 10.8x is approximately x = -0.4587.

Checking the Solution

To make sure we haven't made any mistakes, it's always a good idea to check our solution. We do this by plugging our value of x back into the original equation and seeing if both sides are equal. Let's try it:

Original equation: -17x + 13 + 6x - 23 = 10.8x Substitute x = -0.4587:

• -17(-0.4587) + 13 + 6(-0.4587) - 23 = 10.8(-0.4587)
• 7.8 - 2.75 + 13 - 23 ≈ -4.95
• -4.95 ≈ -4.95

The two sides are approximately equal (there might be a tiny difference due to rounding), so our solution is correct! We can confidently say that x ≈ -0.4587 is the solution to the given linear equation.

Strategies for Solving Linear Equations

Alright, now that we've walked through a specific example, let's zoom out and discuss some general strategies for solving linear equations. These tips and tricks will help you tackle any linear equation that comes your way. The most crucial strategy is to always maintain balance. Remember the seesaw analogy? Whatever operation you perform on one side of the equation, you absolutely must perform the same operation on the other side. This ensures that the equality remains true throughout the solving process. It's like a golden rule for solving equations!

Another essential strategy is to simplify both sides of the equation as much as possible before you start moving terms around. This often involves combining like terms, as we did in our example. Simplifying early on can make the equation much easier to work with and reduce the chances of making errors. Think of it as tidying up your workspace before you start a project; a clean and organized equation is much easier to solve!

Isolate the variable is the name of the game when solving linear equations. Your ultimate goal is to get the variable (usually x) all by itself on one side of the equation. To do this, you'll need to use inverse operations. Inverse operations are those that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. If a term is being added to the variable, subtract it from both sides. If the variable is being multiplied by a number, divide both sides by that number. By systematically applying inverse operations, you can peel away the layers surrounding the variable until it stands alone.

Check your solution, guys! It's a step that's often skipped, but it's super important. Plugging your solution back into the original equation is the best way to catch any mistakes you might have made along the way. If the equation holds true when you substitute your solution, you can be confident that you've got the right answer. If not, you know you need to go back and check your work. It's like proofreading your essay before you submit it; it's a final check to ensure accuracy.

Common Mistakes to Avoid

Solving linear equations is a fundamental skill in algebra, but it's also an area where many students make common mistakes. Let's highlight some of these pitfalls so you can steer clear of them and ace your algebra problems. One of the most frequent errors is forgetting to distribute properly. Distribution is when you multiply a term outside parentheses by each term inside the parentheses. For example, in the expression 2(x + 3), you need to multiply both the x and the 3 by 2, resulting in 2x + 6. Forgetting to distribute to all terms inside the parentheses can lead to a completely wrong answer. So, always double-check that you've distributed correctly!

Another common mistake is combining unlike terms. Remember, like terms have the same variable raised to the same power. You can combine 3x and 5x because they both have x to the power of 1. But you cannot combine 3x and 5x² because one has x to the power of 1 and the other has x to the power of 2. Mixing up like and unlike terms can throw off your entire solution. Always be mindful of the variables and their exponents when combining terms.

Incorrectly applying inverse operations is another pitfall. As we discussed earlier, inverse operations are crucial for isolating the variable. But it's essential to apply the correct inverse operation. If a term is being added, you subtract to undo it. If a term is being multiplied, you divide to undo it. Using the wrong operation can lead you down the wrong path. So, be sure to identify the operation being performed on the variable and use its inverse to undo it.

Finally, not checking your solution is a mistake that can cost you points on exams and assignments. As we emphasized earlier, plugging your solution back into the original equation is a vital step. It's your chance to catch any errors you might have made along the way. Skipping this step is like submitting a report without proofreading it; you might miss a simple mistake that could have been easily corrected. So, always take the time to check your solution and ensure accuracy.

By understanding these common mistakes and actively working to avoid them, you'll be well on your way to mastering linear equations and succeeding in algebra.

Conclusion

So, there you have it! We've successfully solved the linear equation -17x + 13 + 6x - 23 = 10.8x and discussed the key strategies for tackling any linear equation that comes your way. Remember, the key is to simplify, maintain balance, and isolate the variable. And don't forget to check your solution! With practice and a solid understanding of these concepts, you'll be a linear equation whiz in no time. Keep practicing, keep learning, and you'll ace those math problems!