Solving Equations: A Step-by-Step Guide

by SLV Team 40 views

Hey math enthusiasts! Today, we're diving into the world of equations, specifically tackling the problem of solving the equation: 1x−3=1−2x−7x−3\frac{1}{x-3} = 1 - \frac{2x-7}{x-3}. Don't worry if it looks a bit intimidating at first; we'll break it down step by step to make sure you understand every bit of it. Solving equations is a fundamental skill in mathematics, and with a little practice, you'll become a pro at it. Let's get started, shall we?

Understanding the Basics: Equations and Variables

Alright, before we jump into the equation, let's refresh some basics. Equations are mathematical statements that show two expressions are equal. They always have an equals sign (=), which is super important because it shows the balance between the two sides. Think of it like a seesaw; to keep it balanced, whatever you do to one side, you gotta do to the other.

Now, what about the variables? Variables are like placeholders, usually represented by letters (like x in our case). They stand for unknown values that we need to find. The goal of solving an equation is to isolate the variable, meaning we want to get it by itself on one side of the equation. This will give us the value that makes the equation true. Getting comfortable with these concepts is the first step in solving any equation, so take a moment to ensure you have a solid grasp. Remember, practice makes perfect, and solving various equations will help solidify these foundational concepts. The more you solve, the more familiar you become with the process. Feel free to use online equation solvers to check your work, but try doing it on your own first! The sense of accomplishment when you solve an equation independently is a great feeling, and it helps you learn. This is why we are here, right?

We will approach the equation with a series of systematic steps. Each step builds on the previous one, and by the end, we'll have found the solution. Let's start by looking at our equation again: 1x−3=1−2x−7x−3\frac{1}{x-3} = 1 - \frac{2x-7}{x-3}. We are going to solve it together, step by step, so pay close attention.

Step-by-Step Solution: Unraveling the Equation

Step 1: Clearing the Fractions

Okay, guys, the first thing we want to do is get rid of those pesky fractions. Fractions can sometimes make equations look more complicated than they really are, so let's simplify things. To clear the fractions, we need to find the least common denominator (LCD). In our equation, the denominators are x - 3 and 1 (remember, the '1' on the right side of the equation can be thought of as 1/1). The LCD in this case is x - 3, since the other side of the equation has the same denominator. Now, we're going to multiply every term in the equation by (x - 3). Remember, whatever we do to one side, we must do to the other to keep things balanced.

So, let's see how that looks:

  • Original equation: 1x−3=1−2x−7x−3\frac{1}{x-3} = 1 - \frac{2x-7}{x-3}
  • Multiply by (x - 3): (x−3)∗1x−3=(x−3)∗1−(x−3)∗2x−7x−3(x-3) * \frac{1}{x-3} = (x-3) * 1 - (x-3) * \frac{2x-7}{x-3}

Watch carefully how everything simplifies. On the left side, the (x - 3) terms cancel out, leaving us with 1. On the right side, we distribute (x - 3) to both terms. The second term on the right also has the (x - 3) terms cancel out. So, the equation simplifies to:

1 = (x - 3) - (2x - 7)

Awesome, no more fractions! Now it's time to simplify even further.

Step 2: Simplifying and Collecting Like Terms

Now that we've cleared the fractions, let's simplify the equation by expanding and collecting like terms. Expanding means to get rid of the parentheses. We'll do this by distributing the negative sign in front of the (2x - 7). Remember, a negative sign in front of parentheses changes the sign of each term inside the parentheses. So we have

1 = x - 3 - 2x + 7

Next, let's collect the like terms. Like terms are terms that have the same variable raised to the same power. In our equation, the like terms are the x terms (x and -2x) and the constant terms (-3 and +7). Combining the x terms gives us: x - 2x = -x. Combining the constant terms gives us: -3 + 7 = 4. So now our equation looks like this:

1 = -x + 4

We're getting closer to isolating the variable x!

Step 3: Isolating the Variable

Alright, almost there! Now, let's isolate the variable x. This means we want to get the x term by itself on one side of the equation. To do this, we need to get rid of the '+ 4' on the right side. We can do this by subtracting 4 from both sides of the equation. This maintains the balance of the equation.

So, we have:

1 - 4 = -x + 4 - 4

This simplifies to:

-3 = -x

Now, we almost have x isolated. We just need to get rid of the negative sign.

Step 4: Solving for x

We're in the home stretch, guys! To solve for x, we need to get rid of that negative sign in front of the x. The easiest way to do this is to multiply both sides of the equation by -1. Remember, this doesn't change the value of the equation, it simply changes the signs.

So:

-1 * -3 = -1 * -x

This simplifies to:

3 = x

Or, as we usually write it:

x = 3

Checking Your Solution

Before we celebrate, we should always check our solution to make sure we didn't make any mistakes along the way. We do this by substituting the value we found for x back into the original equation. Our original equation was:

1x−3=1−2x−7x−3\frac{1}{x-3} = 1 - \frac{2x-7}{x-3}

Now, substitute x = 3:

13−3=1−2(3)−73−3\frac{1}{3-3} = 1 - \frac{2(3)-7}{3-3}

This simplifies to:

10=1−6−70\frac{1}{0} = 1 - \frac{6-7}{0}

10=1−−10\frac{1}{0} = 1 - \frac{-1}{0}

Whoa, hold on a second! We've run into a bit of a snag. Division by zero is undefined in mathematics. This means that x = 3 is not a valid solution because it makes the denominator of the fractions in the original equation equal to zero. When this happens, we say that the solution is extraneous. The presence of x in the denominator means that there's a restriction on the values x can take. In this case, x cannot equal 3.

So, what does that mean for our equation? It means that there is no solution to the equation. There's no value of x that satisfies the equation because the original equation is not defined when x = 3. Sometimes, after doing all that work, there is no solution!

Conclusion: Mastering Equation Solving

Congratulations, guys! You've successfully navigated a tricky equation, learning about how to clear fractions, collect like terms, isolate variables, and most importantly, how to check your solution. While it might seem a little deflating that there was no solution in this specific case, it's a valuable lesson in understanding the nuances of mathematical equations and the importance of checking your work. This is the process. Remember, the journey of solving an equation involves a series of logical steps, each designed to bring you closer to the solution. Be patient, take your time, and don't be afraid to double-check your work along the way.

Key Takeaways:

  • Clear Fractions: Use the LCD to get rid of fractions.
  • Simplify: Expand and collect like terms.
  • Isolate the Variable: Get the variable by itself.
  • Check Your Solution: Always substitute the solution back into the original equation.
  • Be Aware of Extraneous Solutions: Watch out for values that make the denominator zero.

Keep practicing, and you'll become a pro in no time! Keep up the great work, and remember, mathematics is a skill that develops with consistent effort and a curious mind. Keep practicing with different types of equations, and don't hesitate to ask for help if you get stuck. Embrace the challenge and enjoy the satisfaction of finding solutions! You've got this!