Solving Equations: Step-by-Step Guide For Tomorrow's Class
Hey guys! Let's break down these equations step-by-step so you're totally ready for class tomorrow. We'll go through each one, making sure you understand the process and why we're doing what we're doing. Think of it as unlocking a puzzle – each step gets us closer to the solution!
Understanding the Basics
Before we dive into the specific equations, let's quickly recap the golden rules of equation-solving. The main goal is to isolate the variable (in this case, 'X') on one side of the equation. This means getting 'X' by itself, so we know what its value is. To do this, we use inverse operations. Remember, whatever you do to one side of the equation, you must do to the other side to keep things 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.
Think of inverse operations like this:
- The opposite of addition is subtraction.
- The opposite of subtraction is addition.
- The opposite of multiplication is division.
- The opposite of division is multiplication.
We'll be using these principles throughout the solutions, so keep them in mind. Now, let's tackle those equations!
a) X/3 - 1 = X/6
Okay, let's kick things off with the first equation: X/3 - 1 = X/6. This one involves fractions, which might seem a bit intimidating at first, but don't worry, we'll make it super clear. The first thing we want to do is get rid of those fractions. The easiest way to do that is to find the least common multiple (LCM) of the denominators (the numbers at the bottom of the fractions). In this case, the denominators are 3 and 6. The LCM of 3 and 6 is 6.
So, we're going to multiply both sides of the equation by 6. Remember, whatever we do to one side, we have to do to the other! This gives us:
6 * (X/3 - 1) = 6 * (X/6)
Now, we distribute the 6 on the left side:
(6 * X/3) - (6 * 1) = 6 * (X/6)
This simplifies to:
2X - 6 = X
See how the fractions are gone? Awesome! Now we want to get all the 'X' terms on one side of the equation. Let's subtract 'X' from both sides:
2X - 6 - X = X - X
This gives us:
X - 6 = 0
Almost there! Now, let's isolate 'X' by adding 6 to both sides:
X - 6 + 6 = 0 + 6
And finally, we get our solution:
X = 6
So, the solution to equation (a) is X = 6. Great job!
b) X = 8 + X/5
Let's move on to equation (b): X = 8 + X/5. This one has an 'X' term on both sides, but don't sweat it, we'll handle it like pros. Just like before, we want to get rid of the fraction first. We have a denominator of 5, so let's multiply both sides of the equation by 5:
5 * X = 5 * (8 + X/5)
Distribute the 5 on the right side:
5X = (5 * 8) + (5 * X/5)
This simplifies to:
5X = 40 + X
Now, let's get all the 'X' terms on one side. Subtract 'X' from both sides:
5X - X = 40 + X - X
This gives us:
4X = 40
Almost there! To isolate 'X', we need to divide both sides by 4:
4X / 4 = 40 / 4
And we get our solution:
X = 10
So, the solution to equation (b) is X = 10. You're on fire!
c) X/2 - 3 = -9 - X/4
Now let's tackle equation (c): X/2 - 3 = -9 - X/4. This one has fractions and negative numbers, but don't let that scare you! We'll break it down step-by-step, just like the others. The first step, as you might have guessed, is to get rid of those fractions. We have denominators of 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.
So, we're going to multiply both sides of the equation by 4:
4 * (X/2 - 3) = 4 * (-9 - X/4)
Distribute the 4 on both sides:
(4 * X/2) - (4 * 3) = (4 * -9) - (4 * X/4)
This simplifies to:
2X - 12 = -36 - X
See? The fractions are gone! Now, let's get all the 'X' terms on one side. Add 'X' to both sides:
2X - 12 + X = -36 - X + X
This gives us:
3X - 12 = -36
Next, let's get the constant terms (the numbers without 'X') on the other side. Add 12 to both sides:
3X - 12 + 12 = -36 + 12
This simplifies to:
3X = -24
Finally, to isolate 'X', divide both sides by 3:
3X / 3 = -24 / 3
And we get our solution:
X = -8
So, the solution to equation (c) is X = -8. You're doing amazing!
d) X/3 - 7 = 4 + X/5
Last but not least, let's solve equation (d): X/3 - 7 = 4 + X/5. This equation combines fractions, constants, and 'X' terms on both sides, but we've got this! Just follow the same steps we've been using. First, let's eliminate the fractions. We have denominators of 3 and 5. The least common multiple (LCM) of 3 and 5 is 15.
So, we're going to multiply both sides of the equation by 15:
15 * (X/3 - 7) = 15 * (4 + X/5)
Distribute the 15 on both sides:
(15 * X/3) - (15 * 7) = (15 * 4) + (15 * X/5)
This simplifies to:
5X - 105 = 60 + 3X
Fractions are gone – hooray! Now, let's get all the 'X' terms on one side. Subtract 3X from both sides:
5X - 105 - 3X = 60 + 3X - 3X
This gives us:
2X - 105 = 60
Next, let's get the constant terms on the other side. Add 105 to both sides:
2X - 105 + 105 = 60 + 105
This simplifies to:
2X = 165
Finally, to isolate 'X', divide both sides by 2:
2X / 2 = 165 / 2
And we get our solution:
X = 82.5
So, the solution to equation (d) is X = 82.5. You nailed it!
Key Takeaways & Tips for Success
- LCM is your friend: Finding the least common multiple makes getting rid of fractions way easier.
- Balance is key: Remember to do the same operation on both sides of the equation to keep it balanced.
- Isolate the variable: The goal is to get 'X' by itself on one side of the equation.
- Check your work: Once you've found a solution, plug it back into the original equation to make sure it works.
Final Thoughts
You guys have just tackled some tricky equations! Remember, the more you practice, the easier it gets. If you're still feeling a bit unsure, go back through these examples and try solving them on your own. Understanding the process is much more important than just memorizing the answers. You've got this! Good luck with class tomorrow, and remember, math can be fun!
