Solving Algebraic Equations: Step-by-Step Examples

Hey guys! Today, we're diving into the awesome world of algebraic equations. Don't worry if that sounds intimidating – we're going to break it down into simple, manageable steps. We'll be focusing on solving equations where you need to distribute a number across parentheses. These are super common, and once you get the hang of them, you'll feel like a math whiz! Think of algebraic equations like puzzles. Our goal is to figure out what the mystery number (usually represented by the letter x) is. We do this by using mathematical operations to isolate x on one side of the equation. Let's get started and unlock the secrets of algebra!

In this article, we'll walk through several examples, showing you step-by-step how to solve each one. By the end, you’ll be able to tackle these types of equations with confidence. Remember, the key is to take it one step at a time and stay organized. Keep practicing, and you’ll be solving equations like a pro in no time! We’ll cover equations where you need to distribute a number and then solve for x. This involves using the distributive property, which is a fancy way of saying we multiply the number outside the parentheses by each term inside the parentheses. Once we've done that, it's just a matter of using basic algebraic principles to get x all by itself on one side of the equation. So, grab your pencil and paper, and let's jump in! We're going to learn how to solve equations like: 2(x + 1) = 8, 3(x - 1) = 9, and many more. Each equation is a little challenge, and we're going to conquer them together.

Why is this important? Well, solving algebraic equations is a fundamental skill in mathematics. It’s used in all sorts of fields, from science and engineering to economics and computer programming. Plus, it helps you develop logical thinking and problem-solving skills that are valuable in everyday life. So, let's get started on our algebraic adventure!

Example 1: 2(x + 1) = 8

Let's tackle our first equation: 2(x + 1) = 8. To solve this, we need to isolate x. First, we apply the distributive property. This means we multiply the 2 by both terms inside the parentheses: x and 1. So, 2 * x becomes 2x, and 2 * 1 becomes 2. Our equation now looks like this: 2x + 2 = 8.

The next step is to get the term with x by itself on one side of the equation. To do this, we subtract 2 from both sides. This keeps the equation balanced – what we do to one side, we must do to the other! So, 2x + 2 - 2 = 8 - 2. This simplifies to 2x = 6. Now we're almost there!

Finally, to solve for x, we need to get rid of the 2 that's multiplying it. We do this by dividing both sides of the equation by 2. So, (2x) / 2 = 6 / 2. This simplifies to x = 3. And there you have it! We've solved our first equation. x equals 3. To double-check our answer, we can plug it back into the original equation: 2(3 + 1) = 8. 2 * 4 = 8, which is correct! So we know we’ve got the right answer.

This process of distributing, simplifying, and isolating x is the key to solving these types of equations. Remember, the goal is to keep the equation balanced at each step. So, always perform the same operation on both sides. With practice, these steps will become second nature. Let's move on to another example and build our skills even further!

Example 2: 3(x - 1) = 9

Now, let's move on to our second equation: 3(x - 1) = 9. Just like before, the first thing we need to do is apply the distributive property. This means we multiply the 3 by both terms inside the parentheses: x and -1. So, 3 * x becomes 3x, and 3 * -1 becomes -3. Our equation now looks like this: 3x - 3 = 9.

The next step is to isolate the term with x on one side of the equation. To do this, we add 3 to both sides. Remember, keeping the equation balanced is crucial! So, 3x - 3 + 3 = 9 + 3. This simplifies to 3x = 12. We're getting closer to solving for x!

To finally solve for x, we need to get rid of the 3 that's multiplying it. We do this by dividing both sides of the equation by 3. So, (3x) / 3 = 12 / 3. This simplifies to x = 4. Awesome! We've solved another equation. In this case, x equals 4. To verify our answer, we can substitute it back into the original equation: 3(4 - 1) = 9. 3 * 3 = 9, which is correct! We've nailed it.

Notice how the process is the same as in the first example. Distribute, simplify, and isolate x. The key is to be careful with your signs (positive and negative) and to always perform the same operation on both sides of the equation. Let's continue with more examples to reinforce these steps. Each equation is a chance to practice and build your confidence. So, let's keep going and become equation-solving experts!

Example 3: 5(x + 2) = 15

Let’s tackle our third equation: 5(x + 2) = 15. Just like the previous examples, our first step is to use the distributive property. We multiply the 5 by each term inside the parentheses: x and +2. So, 5 * x becomes 5x, and 5 * 2 becomes 10. This gives us the equation: 5x + 10 = 15.

Now, we need to isolate the term with x. To do this, we subtract 10 from both sides of the equation. Remember, maintaining balance is key! So, we have 5x + 10 - 10 = 15 - 10, which simplifies to 5x = 5.

To finally solve for x, we need to divide both sides of the equation by 5. This will get x all by itself. So, (5x) / 5 = 5 / 5, which simplifies to x = 1. Fantastic! We've solved another equation. Here, x equals 1. Let's check our answer by plugging it back into the original equation: 5(1 + 2) = 15. 5 * 3 = 15, which is correct. Great job!

By now, you might be noticing a pattern. The steps are the same for each equation: distribute, simplify, and isolate x. The more you practice, the more natural these steps will become. Remember to pay close attention to the signs and always keep the equation balanced. Now, let's move on to the next example and keep honing our skills. We're becoming algebraic equation-solving masters!

Example 4: 4(x + 2) = 36

Let's dive into our fourth equation: 4(x + 2) = 36. Just like the previous examples, we start by applying the distributive property. This means we multiply the 4 by both terms inside the parentheses: x and +2. So, 4 * x becomes 4x, and 4 * 2 becomes 8. Our equation now looks like this: 4x + 8 = 36.

The next step is to isolate the term with x on one side of the equation. To do this, we subtract 8 from both sides. Remember, we need to keep the equation balanced! So, 4x + 8 - 8 = 36 - 8. This simplifies to 4x = 28.

Now, to solve for x, we need to get rid of the 4 that's multiplying it. We do this by dividing both sides of the equation by 4. So, (4x) / 4 = 28 / 4. This simplifies to x = 7. Excellent! We've solved another equation. In this case, x equals 7. To double-check, we plug our answer back into the original equation: 4(7 + 2) = 36. 4 * 9 = 36, which is correct. We're on a roll!

Each time we solve an equation, we're reinforcing the process and building our confidence. The steps are consistent: distribute, simplify, and isolate x. Pay attention to the details, like signs and keeping the equation balanced, and you'll be successful every time. Let's keep the momentum going and tackle another example. We're getting closer to mastering these equations!

Example 5: 7(x - 2) = 21

Now, let's tackle equation number five: 7(x - 2) = 21. As with our previous examples, the first step is to apply the distributive property. This means multiplying the 7 by both terms inside the parentheses: x and -2. So, 7 * x becomes 7x, and 7 * -2 becomes -14. This gives us the equation: 7x - 14 = 21.

Next, we need to isolate the term with x on one side of the equation. To do this, we add 14 to both sides. Remember, it's essential to keep the equation balanced! So, 7x - 14 + 14 = 21 + 14. This simplifies to 7x = 35.

To finally solve for x, we need to divide both sides of the equation by 7. This will isolate x. So, (7x) / 7 = 35 / 7, which simplifies to x = 5. Wonderful! We've solved another equation. In this case, x equals 5. To verify our solution, let's substitute it back into the original equation: 7(5 - 2) = 21. 7 * 3 = 21, which is indeed correct. Fantastic work!

You're likely noticing by now that the process is becoming more familiar. We consistently use the same steps: distribute, simplify, and isolate x. With each equation we solve, we're solidifying our understanding and building our skills. Remember to always double-check your answers by plugging them back into the original equation. Now, let's move on to the next example and continue our journey to algebraic mastery. We're doing great!

Example 6: 2(2x + 1) = 26

Moving on to our sixth equation, we have 2(2x + 1) = 26. As before, we start with the distributive property. We multiply the 2 by both terms inside the parentheses: 2x and +1. So, 2 * 2x becomes 4x, and 2 * 1 becomes 2. This gives us the equation: 4x + 2 = 26.

The next step is to isolate the term with x. To do this, we subtract 2 from both sides of the equation. Keeping the equation balanced is crucial! So, 4x + 2 - 2 = 26 - 2. This simplifies to 4x = 24.

To finally solve for x, we need to divide both sides of the equation by 4. This will get x by itself. So, (4x) / 4 = 24 / 4, which simplifies to x = 6. Excellent! We've solved another equation. Here, x equals 6. Let's check our work by plugging it back into the original equation: 2(2 * 6 + 1) = 26. 2(12 + 1) = 2 * 13 = 26, which is correct. Well done!

This example reinforces the importance of careful multiplication and following the same steps consistently. Distribute, simplify, and isolate x. Remember to double-check your work by plugging your solution back into the original equation. Let's keep building our skills with more examples. We're becoming true algebraic equation solvers!

Example 7: 3(2x - 1) = 27

Let's jump into our seventh equation: 3(2x - 1) = 27. As you've probably guessed, we start by applying the distributive property. We multiply the 3 by both terms inside the parentheses: 2x and -1. So, 3 * 2x becomes 6x, and 3 * -1 becomes -3. This gives us the equation: 6x - 3 = 27.

The next step is to isolate the term with x on one side of the equation. To do this, we add 3 to both sides. Keeping the balance of the equation is crucial! So, 6x - 3 + 3 = 27 + 3. This simplifies to 6x = 30.

To finally solve for x, we need to divide both sides of the equation by 6. This will isolate x. So, (6x) / 6 = 30 / 6, which simplifies to x = 5. Wonderful! We've solved yet another equation. In this case, x equals 5. Let's verify our solution by substituting it back into the original equation: 3(2 * 5 - 1) = 27. 3(10 - 1) = 3 * 9 = 27, which is correct. Great job!

With each equation we solve, the process becomes more ingrained in our minds. We're consistently applying the same steps: distribute, simplify, and isolate x. Always remember to double-check your answers by plugging them back into the original equation. Let's keep the momentum going and move on to the next example. We're well on our way to mastering these types of equations!

Example 8: 2(5x + 4) = 28

Now, let's dive into our eighth equation: 2(5x + 4) = 28. As always, we begin by applying the distributive property. We multiply the 2 by both terms inside the parentheses: 5x and +4. So, 2 * 5x becomes 10x, and 2 * 4 becomes 8. This gives us the equation: 10x + 8 = 28.

Our next step is to isolate the term with x on one side of the equation. To do this, we subtract 8 from both sides. It's essential to keep the equation balanced! So, 10x + 8 - 8 = 28 - 8. This simplifies to 10x = 20.

To finally solve for x, we need to divide both sides of the equation by 10. This will isolate x. So, (10x) / 10 = 20 / 10, which simplifies to x = 2. Fantastic! We've solved another equation successfully. In this case, x equals 2. To verify our answer, let's substitute it back into the original equation: 2(5 * 2 + 4) = 28. 2(10 + 4) = 2 * 14 = 28, which is correct. Excellent work!

We're continuing to reinforce our understanding of the process: distribute, simplify, and isolate x. Each time we solve an equation, we're building our confidence and skills. Remember to always double-check your solutions by plugging them back into the original equation. Let's keep practicing with another example and continue our journey to algebraic mastery. We're doing a great job!

Example 9: 3(3x - 7) = 15

Let's tackle our final equation: 3(3x - 7) = 15. As with all the previous examples, we start by applying the distributive property. We multiply the 3 by both terms inside the parentheses: 3x and -7. So, 3 * 3x becomes 9x, and 3 * -7 becomes -21. This gives us the equation: 9x - 21 = 15.

Our next step is to isolate the term with x on one side of the equation. To do this, we add 21 to both sides. Remember, maintaining balance in the equation is crucial! So, 9x - 21 + 21 = 15 + 21. This simplifies to 9x = 36.

To finally solve for x, we need to divide both sides of the equation by 9. This will isolate x. So, (9x) / 9 = 36 / 9, which simplifies to x = 4. Incredible! We've solved our final equation in this set. In this case, x equals 4. To ensure our solution is correct, let's substitute it back into the original equation: 3(3 * 4 - 7) = 15. 3(12 - 7) = 3 * 5 = 15, which is indeed correct. Fantastic!

We've now successfully solved a variety of equations using the same fundamental process: distribute, simplify, and isolate x. This consistent approach is key to mastering these types of algebraic problems. Always remember to double-check your answers by plugging them back into the original equation. You've come a long way in building your algebraic skills, and you should be proud of your progress. Keep practicing, and you'll continue to improve your equation-solving abilities.

Conclusion

Alright guys, we've reached the end of our journey into solving basic algebraic equations! We've covered a lot of ground, from understanding the distributive property to isolating x and verifying our solutions. Remember, the key to success in algebra is practice, practice, practice!

We started by breaking down the basic principles of solving equations. We emphasized the importance of keeping the equation balanced by performing the same operations on both sides. We then walked through nine different examples, each one building on the previous one. We saw how to apply the distributive property, combine like terms, and isolate the variable x. We also stressed the importance of checking your answers by plugging them back into the original equation. This not only confirms your solution but also helps solidify your understanding of the process.

Solving algebraic equations is a crucial skill that you'll use throughout your mathematical journey and beyond. It's not just about finding the right answer; it's about developing logical thinking and problem-solving skills that are valuable in all areas of life. So, keep practicing, keep challenging yourself, and don't be afraid to ask for help when you need it. You've got this! Whether you're tackling more complex equations in the future or using these skills in other fields, the foundation you've built here will serve you well. Keep up the great work, and remember, every equation you solve is a step towards mastering algebra! Congratulations on taking this step in your math education.