Unlocking Equations: A Step-by-Step Guide

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Unlocking Equations: A Step-by-Step Guide to Solving Mathematical Problems

Hey math enthusiasts! Ever feel like equations are these mysterious puzzles you just can't crack? Well, fret not! Solving equations is like learning a secret code that unlocks a whole new world of problem-solving. In this guide, we're diving deep into some common equation types, breaking them down step by step, and making sure you feel confident in your equation-solving abilities. We'll be tackling several equations, each designed to illustrate different aspects of the process. So, grab your pencils, and let's get started!

Equation 1: 28 - 7(x - 11) = 0

Okay, guys, let's start with our first equation: 28 - 7(x - 11) = 0. This one involves parentheses, which means we'll need to use the distributive property first. Remember, the distributive property means multiplying the number outside the parentheses by each term inside. Here's how we'll break it down:

  1. Distribute the -7: Multiply -7 by both x and -11. This gives us: 28 - 7x + 77 = 0.
  2. Combine like terms: Add 28 and 77, which gives us: 105 - 7x = 0.
  3. Isolate the variable term: Subtract 105 from both sides of the equation: -7x = -105.
  4. Solve for x: Divide both sides by -7: x = 15.

So, the solution to the equation 28 - 7(x - 11) = 0 is x = 15. See? Not so scary, right? Always remember to follow the order of operations (PEMDAS/BODMAS) to ensure you're on the right track. This equation is a classic example of how to handle parentheses and combine like terms. This initial step is critical, as it sets the foundation for all subsequent calculations. Ensuring accuracy at this stage minimizes errors later. Now we know, using the distributive property simplifies the equation and makes it easier to manage. Remember to carefully consider the signs when distributing. A common mistake is forgetting to distribute the negative sign, so always pay close attention.

Let's make sure that our solution is correct. We can substitute x=15 back into the original equation: 28-7(15-11)=0. Which is 28-7(4)=0, which is 28-28=0. Our calculation is correct. Remember, verifying your solution is a great way to build your confidence and catch any mistakes you might have made.

Equation 2: 9(x - 7) - 38 = 7

Alright, let's move on to the next challenge: 9(x - 7) - 38 = 7. This equation is quite similar to the last one, as we'll need to use the distributive property again. Let's walk through it together:

  1. Distribute the 9: Multiply 9 by both x and -7. This gives us: 9x - 63 - 38 = 7.
  2. Combine like terms: Combine -63 and -38: 9x - 101 = 7.
  3. Isolate the variable term: Add 101 to both sides: 9x = 108.
  4. Solve for x: Divide both sides by 9: x = 12.

So, the solution to the equation 9(x - 7) - 38 = 7 is x = 12. Remember, guys, the key here is to stay organized and take things one step at a time. Do not rush the steps, and double-check your calculations to prevent silly mistakes. The more you practice, the easier it becomes.

Let's verify the solution by plugging x=12 into the original equation: 9(12-7)-38=7. This is 9(5)-38=7. Which is 45-38=7. Our result is correct. Keep up the good work; you're doing great. Always ensure that you are following the order of operations (PEMDAS/BODMAS), which guarantees accuracy.

Equation 3: 4(x - 8) - 23 = 1

Now, let's solve 4(x - 8) - 23 = 1. This is very similar to the last couple of equations, so you're probably getting the hang of it now! Let's get started:

  1. Distribute the 4: Multiply 4 by both x and -8. This gives us: 4x - 32 - 23 = 1.
  2. Combine like terms: Combine -32 and -23: 4x - 55 = 1.
  3. Isolate the variable term: Add 55 to both sides: 4x = 56.
  4. Solve for x: Divide both sides by 4: x = 14.

Therefore, the solution to 4(x - 8) - 23 = 1 is x = 14. You guys are rockstars! Let's continue building up our equation-solving skills. Remember that the distributive property is one of the most important tools in solving this type of equation. It allows us to remove the parentheses and make the equation more manageable. After distributing, combining like terms is important for simplification, and isolating the variable term is necessary to get closer to the final solution.

We can plug the value x=14 to confirm that we are correct: 4(14-8)-23=1. 4(6)-23=1, which is 24-23=1. Hence, the solution is correct.

Equation 4: 21 + 3(3 - x) = 30

Okay, let's mix things up a bit with 21 + 3(3 - x) = 30. This time, the negative variable is inside the parenthesis. Don't worry, it's still manageable! Here's how to solve it:

  1. Distribute the 3: Multiply 3 by both 3 and -x. This gives us: 21 + 9 - 3x = 30.
  2. Combine like terms: Combine 21 and 9: 30 - 3x = 30.
  3. Isolate the variable term: Subtract 30 from both sides: -3x = 0.
  4. Solve for x: Divide both sides by -3: x = 0.

So, the solution to the equation 21 + 3(3 - x) = 30 is x = 0. Notice how the negative sign in front of the variable can sometimes trip us up. Be mindful of those details. Keep in mind that negative signs are critical. When simplifying equations, they can change the entire result. Double-check your calculations to ensure accuracy.

Let's check our solution: 21+3(3-0)=30. Which is 21+3(3)=30, 21+9=30. Our solution is verified.

Equation 5: 4(x - 6) + 12 = 52

Let's move on to 4(x - 6) + 12 = 52. This equation is similar to those we've already solved, so you're probably feeling like a pro by now! Let's solve it together:

  1. Distribute the 4: Multiply 4 by both x and -6. This gives us: 4x - 24 + 12 = 52.
  2. Combine like terms: Combine -24 and 12: 4x - 12 = 52.
  3. Isolate the variable term: Add 12 to both sides: 4x = 64.
  4. Solve for x: Divide both sides by 4: x = 16.

Therefore, the solution to the equation 4(x - 6) + 12 = 52 is x = 16. As you continue to practice, you'll become more efficient at these steps. With each equation you solve, you're becoming more and more confident. The more you practice, the easier and faster it becomes. Keep up the great work!

Verify that our answer is correct by substituting x=16 in the original equation: 4(16-6)+12=52. This is 4(10)+12=52, which is 40+12=52. Which makes our calculation correct.

Equation 6: 5(x - 2) + 8 = 108

Alright, let's solve 5(x - 2) + 8 = 108. We're almost done, guys! Let's finish strong:

  1. Distribute the 5: Multiply 5 by both x and -2. This gives us: 5x - 10 + 8 = 108.
  2. Combine like terms: Combine -10 and 8: 5x - 2 = 108.
  3. Isolate the variable term: Add 2 to both sides: 5x = 110.
  4. Solve for x: Divide both sides by 5: x = 22.

So, the solution to the equation 5(x - 2) + 8 = 108 is x = 22. As we advance through these exercises, you will probably notice recurring patterns in the calculations. This allows you to simplify steps mentally and arrive at the solution more quickly. The more you work with equations, the more familiar you will be with the process, which is very helpful in complex equations.

To ensure our solution is correct, we will substitute x=22 into the original equation: 5(22-2)+8=108. 5(20)+8=108 which means 100+8=108. Our solution is verified.

Equation 7: 9(x + 4) - 30 = 6

Finally, let's wrap things up with 9(x + 4) - 30 = 6. This equation is a bit different because it involves addition inside the parentheses, but the process is still the same. Let's solve it together:

  1. Distribute the 9: Multiply 9 by both x and 4. This gives us: 9x + 36 - 30 = 6.
  2. Combine like terms: Combine 36 and -30: 9x + 6 = 6.
  3. Isolate the variable term: Subtract 6 from both sides: 9x = 0.
  4. Solve for x: Divide both sides by 9: x = 0.

Therefore, the solution to the equation 9(x + 4) - 30 = 6 is x = 0. Great job completing all the equations! Remember, guys, practice makes perfect. The more you solve equations, the more comfortable and confident you will become. Don't be afraid to make mistakes; they are a part of the learning process. You're doing awesome!

To make sure we are correct, let us put the value x=0 into the original equation: 9(0+4)-30=6. This results in 9(4)-30=6. Which is 36-30=6. Our calculation is correct.

Conclusion

Awesome work, everyone! You've successfully navigated through seven different equations, each highlighting key steps in the solving process. From using the distributive property to isolating variables, you've gained valuable skills. Remember to always double-check your work and to stay organized. Keep practicing, and you'll be solving equations like a pro in no time! Remember, solving equations is like learning a language – the more you use it, the better you become. Keep practicing, and you'll see your skills improve. Until next time, keep those equations coming!