Solving Equations: Step-by-Step Calculations

Hey math enthusiasts! Ready to dive into the world of equations? Today, we're going to tackle some fundamental algebraic problems, breaking down each step to make sure you understand how to solve them. We'll be working through various equations, from simple addition and subtraction to more complex expressions involving multiplication and variables. Let's get started and make math fun!

Equation a) x + 6 = 10

Alright, first up, we have x + 6 = 10. The goal here is to isolate x, meaning we want to get x all by itself on one side of the equation. To do this, we need to get rid of the + 6. The key principle in solving equations is to do the same thing to both sides. So, since we have + 6 on the left side, we're going to subtract 6 from both sides of the equation. This maintains the balance.

So, we subtract 6 from both sides: x + 6 - 6 = 10 - 6. On the left side, + 6 - 6 cancels out, leaving us with just x. On the right side, 10 - 6 equals 4. Therefore, the solution is x = 4. We've successfully solved our first equation! Always remember, the core concept is to isolate the variable by performing inverse operations on both sides. This method guarantees that the equation remains balanced and true, allowing us to find the correct value for the unknown variable. Understanding this foundational principle is key to tackling more complex algebraic problems down the line, ensuring you maintain a solid grasp of mathematical concepts.

Equation b) x - 3 = 12

Let's move on to the next one: x - 3 = 12. Here, we have x - 3. To get x alone, we need to eliminate the - 3. The inverse operation of subtraction is addition, so we'll add 3 to both sides of the equation. This operation cancels out the -3 on the left side, isolating the variable x.

Adding 3 to both sides gives us: x - 3 + 3 = 12 + 3. On the left side, - 3 + 3 cancels out, leaving just x. On the right side, 12 + 3 equals 15. Thus, we find that x = 15. Notice that we're consistently applying the principle of keeping the equation balanced by performing the same operation on both sides. This ensures that the equality remains valid. Always double-check your arithmetic, as a small mistake can lead to an incorrect answer. With practice, these steps become second nature, and solving equations will become a breeze! Remember, the goal is always to isolate the variable, and the method is to use inverse operations.

Equation c) 3x + 2 = 17

Okay, let's bump up the challenge a bit with 3x + 2 = 17. In this equation, we've got a coefficient (the number multiplying x) and a constant. First, we need to get rid of the + 2. Just like before, we'll subtract 2 from both sides of the equation to isolate the term with x. This prepares us for the next step, where we'll deal with the coefficient.

Subtracting 2 from both sides, we get: 3x + 2 - 2 = 17 - 2. This simplifies to 3x = 15. Now, we have 3x, which means 3 multiplied by x. To isolate x, we need to do the inverse operation of multiplication, which is division. We'll divide both sides of the equation by 3. Doing so isolates x.

Dividing both sides by 3, we have: (3x) / 3 = 15 / 3. This simplifies to x = 5. Therefore, the solution for this equation is x = 5. You'll notice that the process involves multiple steps, but the core principle of maintaining balance through inverse operations remains consistent. Breaking down the problem into smaller, manageable steps is a great strategy for tackling more complex equations. Keep practicing, and you'll find these multi-step problems become increasingly easier to solve.

Equation d) 6x - 24 = 36

Alright, let's solve 6x - 24 = 36. Our first move here is to isolate the 6x term. Because we have – 24, we need to do the opposite and add 24 to both sides of the equation. Adding 24 to both sides helps eliminate the constant on the same side as the x term, bringing us one step closer to isolating the variable.

Adding 24 to both sides: 6x - 24 + 24 = 36 + 24, simplifies to 6x = 60. Now, to solve for x, we must get rid of the coefficient 6. As 6 is multiplying x, we divide both sides by 6. This isolates x, and gives us the solution.

So, divide both sides by 6: (6x) / 6 = 60 / 6. Simplifying gives us x = 10. There you have it! The solution to this equation is x = 10. As you can see, each equation has a systematic approach. By carefully applying the inverse operations and maintaining balance, we can solve for x with confidence. Remember to double-check your calculations, especially when dealing with larger numbers or multiple steps.

Equation e) 3x + x = 24

Next, let’s solve 3x + x = 24. This one is a bit different because we have like terms on the same side of the equation. Our first step is to combine these like terms. The 3x and x can be combined, which simplifies the equation considerably.

Combining the like terms, 3x + x is the same as 4x. So, our equation becomes 4x = 24. Now, we need to isolate x. Since 4 is multiplied by x, we divide both sides by 4 to get the value of x.

Divide both sides by 4: (4x) / 4 = 24 / 4. Simplifying, we get x = 6. And there you have the solution! For this equation, x equals 6. Simplifying before you start is often the key to making an equation easier to solve. When you can combine like terms, it makes the later steps of isolating the variable much simpler. Practice with these types of equations will help you quickly recognize the best approach.

Equation f) 3x + x - 7 = 21

Now, let's work on 3x + x - 7 = 21. This equation requires us to first combine like terms on the same side of the equation, as we saw in the previous example. Then, we will isolate the variable x using inverse operations.

Let’s combine the like terms, 3x + x simplifies to 4x. So we now have 4x - 7 = 21. Next, we'll isolate the term with the variable by adding 7 to both sides of the equation. This cancels out the – 7 on the left side, moving us towards isolating x.

Adding 7 to both sides, we get 4x - 7 + 7 = 21 + 7. This simplifies to 4x = 28. Now, we divide both sides by 4 to solve for x: (4x) / 4 = 28 / 4. That gives us x = 7. There you have it! For this equation, x is equal to 7. Remember to be meticulous with the order of operations and to check your answers when solving any equation. This will ensure you’re always accurate.

Equation g) 7x - 2 = 22 + 3x

Let’s tackle 7x - 2 = 22 + 3x. This one is a bit more complex since we have x terms on both sides of the equation. Our first step is to get all the x terms on one side. We can do this by subtracting 3x from both sides of the equation. This eliminates the 3x term from the right side, but we must also perform this operation on the left to maintain balance.

Subtracting 3x from both sides gives us 7x - 3x - 2 = 22 + 3x - 3x. This simplifies to 4x - 2 = 22. Now, we must isolate the x term. The next step is to add 2 to both sides of the equation, which cancels out the -2 and gets the x term by itself.

Adding 2 to both sides, we have 4x - 2 + 2 = 22 + 2, which simplifies to 4x = 24. Finally, divide both sides by 4: (4x) / 4 = 24 / 4. This gives us x = 6. Thus, the solution is x = 6. Dealing with x terms on both sides of the equation requires a few extra steps, but the fundamental principles remain the same. The key is to consolidate like terms and isolate the variable.

Equation h) 12 + 3x = 28 + x

Alright, let's solve the last equation, 12 + 3x = 28 + x. Like the previous example, we have x terms on both sides. Our first move is to collect the x terms on one side. Subtracting x from both sides will help us consolidate the x terms.

Subtracting x from both sides results in: 12 + 3x - x = 28 + x - x. This simplifies to 12 + 2x = 28. Now, we want to isolate the term with x. We do this by subtracting 12 from both sides of the equation to isolate the term with x.

Subtracting 12 from both sides gives us: 12 - 12 + 2x = 28 - 12. Which simplifies to 2x = 16. Finally, we can isolate x by dividing both sides by 2: (2x) / 2 = 16 / 2. This simplifies to x = 8. The answer is x = 8. Congrats, you've worked through a variety of equation-solving scenarios! Remember to take your time, double-check your work, and always aim to understand the underlying principles.

Well done, guys! You've successfully worked through a variety of equations, and hopefully, you now have a solid understanding of how to solve them. Keep practicing, and you'll become more confident and proficient in your equation-solving skills. Math can be tricky, but with the right approach and a little patience, you can master it! Keep up the great work, and happy solving! If you enjoyed this and want more math practice, feel free to ask!