Solve For X: Step-by-Step Guide
Hey guys! Today, we're diving into a classic algebra problem: solving for x. Don't worry if equations sometimes look like a jumbled mess; we'll break it down step by step so it's super clear. Let's tackle this equation together:
We're going to simplify it and find out exactly what x equals. Ready? Let's jump in!
1. Understanding the Equation: The Foundation of Solving for x
When you first look at an algebraic equation, like our example, , it might seem a bit daunting. But trust me, understanding the anatomy of the equation is the crucial first step. The primary goal in solving for x is to isolate x on one side of the equation. This means we want to manipulate the equation using mathematical operations until we have x all by itself, with its value clearly stated on the other side. Think of it like peeling away layers to reveal the treasure hidden inside – in this case, the value of x.
Our equation has several key components. We have a fraction on the left side, which includes the variable x. The expression 5x - 17 is the numerator, and -3 is the denominator. On the right side, we have a simple integer, -6. The equal sign acts as a balance, showing that whatever is on the left side has the same value as whatever is on the right side. To successfully solve for x, we need to maintain this balance while we simplify the equation.
The operations we perform on the equation must be done on both sides to keep it balanced. It's like a seesaw – if you add or remove weight from one side, you must do the same on the other to keep it level. Common operations include addition, subtraction, multiplication, and division. We'll use these operations strategically to undo what's being done to x, gradually isolating it. Understanding the order of operations (PEMDAS/BODMAS) is also crucial. We'll often work in reverse, undoing addition and subtraction before multiplication and division.
In this specific equation, we see that x is part of an expression that's being divided by -3. To start isolating x, we'll need to address this division first. By understanding the structure of the equation and the principles of algebraic manipulation, we set ourselves up for a clear and methodical solution. So, let's move on to the next step: eliminating the fraction.
2. Eliminating the Fraction: Clearing the Path to x
The next step in solving our equation, , is to eliminate the fraction. Fractions can sometimes make equations look more complicated than they are, so getting rid of them often simplifies the process. In our case, we have the entire expression 5x - 17 divided by -3. To undo this division, we need to perform the inverse operation, which is multiplication. We're going to multiply both sides of the equation by -3. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance.
Multiplying both sides by -3 gives us:
On the left side, the -3 in the numerator and the -3 in the denominator cancel each other out. This is the magic of inverse operations at work! We're left with just 5x - 17. On the right side, we have -6 multiplied by -3. A negative times a negative is a positive, so -6 \times -3 equals 18. Our equation now looks much simpler:
See how much cleaner that looks? By eliminating the fraction, we've cleared a significant hurdle in solving for x. Now, x is part of a simpler expression, and we're closer to isolating it. This step highlights the importance of using inverse operations to undo what's being done to the variable. In this case, we used multiplication to undo division. This principle will guide us as we continue to solve the equation.
With the fraction gone, we can focus on the remaining operations affecting x. We have 5x - 17 = 18. The next step involves dealing with the subtraction. So, let's move on and see how we can isolate x further.
3. Isolating the Term with x: Getting Closer to the Solution
We've made great progress! Our equation is now simplified to 5x - 17 = 18. The goal, as always, is to isolate x. Currently, we have x multiplied by 5, and then 17 is subtracted from the result. To isolate the term with x (which is 5x), we need to deal with the subtraction first. We'll use the inverse operation of subtraction, which is addition. We're going to add 17 to both sides of the equation to maintain balance.
Adding 17 to both sides gives us:
On the left side, the -17 and +17 cancel each other out. This is exactly what we wanted! We're left with just 5x. On the right side, 18 + 17 equals 35. Our equation now looks like this:
We're getting so close! We've successfully isolated the term containing x. Now, x is only being multiplied by 5. This is a major step forward in our solution. By adding 17 to both sides, we effectively undid the subtraction and moved the constant term to the right side of the equation. This highlights the importance of performing inverse operations strategically. Each step brings us closer to having x all by itself.
With 5x = 35, we have one final operation to undo: the multiplication. The next step will involve using division to isolate x completely. So, let's move on to the final step and find the value of x.
4. Solving for x: The Final Showdown
We've reached the final stretch! Our equation is 5x = 35. We are now at the pivotal moment where we isolate x completely. Currently, x is being multiplied by 5. To undo this multiplication and get x by itself, we need to perform the inverse operation: division. We're going to divide both sides of the equation by 5.
Dividing both sides by 5 gives us:
On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with just x. On the right side, 35 divided by 5 equals 7. Our equation now reads:
And there you have it! We've successfully solved for x. The value of x that makes the original equation true is 7. This final step demonstrates the power of using inverse operations to isolate the variable. By dividing both sides by 5, we undid the multiplication and revealed the solution. This is the culmination of all our efforts, and it's a satisfying moment when the value of x is finally revealed.
To ensure our solution is correct, we can always substitute x = 7 back into the original equation and check if both sides are equal. This is a good practice to catch any errors that might have occurred during the solving process. So, let's do that now.
5. Checking the Solution: Ensuring Accuracy
Before we declare victory, it's always a good idea to check our solution. We found that x = 7, but let's make sure this value actually works in the original equation. Plugging our solution back into the equation helps us catch any mistakes we might have made along the way. Our original equation was:
Now, we'll substitute x with 7:
First, we perform the multiplication inside the parentheses:
Next, we subtract 17 from 35:
Finally, we divide 18 by -3:
Lo and behold! The left side of the equation equals the right side. This confirms that our solution, x = 7, is indeed correct. Checking our solution provides peace of mind and ensures that we've solved the equation accurately. It's a simple yet powerful step in the problem-solving process.
Conclusion: Mastering the Art of Solving for x
Great job, guys! We've successfully navigated the equation and found that x = 7. We started by understanding the equation, then systematically eliminated the fraction, isolated the term with x, and finally solved for x. We even took the time to check our solution, ensuring accuracy. This step-by-step approach is the key to tackling any algebraic equation.
Solving for x is a fundamental skill in algebra, and it's a building block for more advanced mathematical concepts. By mastering these basic techniques, you'll be well-equipped to handle a wide range of problems. Remember, the key is to break down complex equations into smaller, manageable steps. Use inverse operations to undo what's being done to x, and always double-check your work.
So, the next time you encounter an equation, don't be intimidated. Remember the steps we've covered here, and you'll be solving for x like a pro in no time! Keep practicing, and you'll become more confident and proficient in your algebra skills. You got this!
Final Answer:
