Solving For X: A Step-by-Step Guide

Have you ever stumbled upon an equation and felt a little intimidated? Don't worry, it happens to the best of us! Equations might seem complex at first glance, but with the right approach, they can be broken down into manageable steps. In this article, we'll walk through the process of solving for x in the equation 7x - 3 = -9(x - 5). We'll break down each step, explain the reasoning behind it, and make sure you understand the underlying principles. So, grab a pen and paper, and let's dive in!

Understanding the Basics of Equations

Before we jump into the specifics of this equation, let's quickly review some fundamental concepts about equations. At its heart, an equation is a statement that two expressions are equal. Think of it like a balanced scale: whatever is on one side must be equal in weight to what's on the other side. Our goal when solving for x is to isolate x on one side of the equation, effectively figuring out what value of x makes the equation true. We achieve this by performing the same operations on both sides of the equation, maintaining that crucial balance.

The golden rule of algebra is that whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain equality. This ensures that the 'balance' of the equation is preserved, and we're one step closer to isolating our variable, x.

Step 1: Distribute the -9

Our equation is 7x - 3 = -9(x - 5). The first thing we need to do is simplify the right side of the equation. Notice the -9 outside the parentheses? This means we need to distribute the -9 to both terms inside the parentheses. Remember, distribution means multiplying the term outside the parentheses by each term inside. So, -9 multiplied by x is -9x, and -9 multiplied by -5 is +45 (a negative times a negative is a positive!).

Our equation now looks like this: 7x - 3 = -9x + 45. We've successfully eliminated the parentheses and made our equation a little easier to work with. This distribution step is a crucial part of simplifying algebraic expressions and paving the way for isolating our target variable, x. By carefully applying the distributive property, we maintain the equation's balance while rearranging terms to facilitate solving.

Step 2: Combine x Terms

Now, we want to get all the x terms on one side of the equation. Looking at our current equation, 7x - 3 = -9x + 45, we have x terms on both sides. It doesn't matter which side we choose, but let's move the -9x term from the right side to the left side. To do this, we add 9x to both sides of the equation (remember the golden rule!). Adding 9x to both sides cancels out the -9x on the right side, leaving us with just the constant term.

This gives us: 7x + 9x - 3 = 45. Now, we can combine the x terms on the left side: 7x + 9x equals 16x. So, our equation simplifies to 16x - 3 = 45. We're making progress! Combining like terms is a fundamental algebraic technique that helps simplify equations and make them easier to solve. By strategically adding or subtracting terms from both sides, we can consolidate variables and constants, bringing us closer to isolating x.

Step 3: Isolate the x Term

Our goal is to get x all by itself on one side of the equation. We're currently at 16x - 3 = 45. The next step is to get rid of the -3 on the left side. To do this, we add 3 to both sides of the equation. This will cancel out the -3 on the left side and leave us with just the term containing x.

Adding 3 to both sides, we get: 16x = 48. Now we have 16x isolated on the left side, and a constant term on the right. We're almost there! This process of isolating the variable term is a critical step in solving equations. By strategically adding or subtracting constants, we can progressively peel away the layers surrounding x, bringing us closer to its value.

Step 4: Solve for x

We're in the home stretch! Our equation is now 16x = 48. To finally solve for x, we need to get rid of the 16 that's multiplying x. We do this by dividing both sides of the equation by 16. This is the inverse operation of multiplication, and it will isolate x on the left side.

Dividing both sides by 16, we get: x = 48 / 16. Now, we simply perform the division: 48 divided by 16 is 3. Therefore, x = 3. We've found our solution! This final step of isolating x by using division (or multiplication, depending on the equation) is the culmination of our efforts. It's where we unveil the specific value that satisfies the equation.

Step 5: Verification

Let's double-check if our solution is correct. To verify, we substitute the value of x we obtained back into the original equation, we have: 7x - 3 = -9(x - 5).

Substitute x = 3, then we have 7(3) - 3 = -9(3 - 5).

Simplify the equation we have, 21 - 3 = -9(-2).

Thus, 18 = 18.

Therefore, our solution is verified to be correct.

Conclusion: Mastering the Art of Solving Equations

And there you have it! We've successfully solved for x in the equation 7x - 3 = -9(x - 5). By following these steps – distributing, combining like terms, isolating the x term, and finally solving for x – you can tackle a wide range of algebraic equations. Remember, practice makes perfect, so don't be afraid to try solving different equations to build your skills and confidence.

Solving equations is a fundamental skill in mathematics and has applications in various fields, from science and engineering to finance and economics. The ability to manipulate equations and isolate variables allows us to model real-world problems, make predictions, and gain a deeper understanding of the relationships between different quantities. So, keep practicing, keep exploring, and you'll find that the world of equations becomes less daunting and more fascinating.

If you want to explore more about equations and algebra, check out resources like Khan Academy's Algebra Section. Happy solving!