Solve Equations: Step-by-Step Guide

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Solve Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of equations. Don't worry, it's not as scary as it sounds. We're going to break down how to solve them step-by-step, making it super easy to understand. So, grab your pencils and let's get started. We'll be solving some basic equations together, like a x-3=5, b x-0,5 = 1,3, and more. By the end, you'll be a pro at finding those missing values. Let's start with some foundational knowledge. Understanding the basics is key to mastering more complex problems. Remember that an equation is like a balanced scale; whatever you do to one side, you have to do to the other to keep it balanced. This fundamental concept underpins all equation-solving techniques. We'll be using this principle throughout our examples. The goal of solving an equation is always to isolate the variable, which means getting it all by itself on one side of the equation. This is achieved by performing inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. The basic structure of any equation usually involves a variable (like x or y), numbers, and mathematical operations (like +, -, *, /). Our goal is to manipulate the equation using these inverse operations until the variable stands alone. Pay close attention to this as we work through the examples. It's the most important principle for success. This first set of equations involves simple arithmetic manipulations. We'll add or subtract numbers from both sides to isolate our variables. Keep in mind, the ultimate goal is to get the variable by itself. This means we'll perform inverse operations to cancel out numbers that are combined with the variable. The process might seem simple, but mastering these basics is crucial before moving to more advanced topics. Feel free to re-read and practice these examples until you're completely comfortable. Remember, practice makes perfect. Now, let’s get our hands dirty!

Solving for Unknown Variables: Examples and Explanations

Alright, let's jump right into some examples. We'll tackle each equation step-by-step, explaining the reasoning behind each move. This approach will help you understand not just what to do, but why you're doing it. By understanding the 'why,' you'll find it easier to solve different types of equations. We're going to work through each problem, explaining the process in detail. Don't worry if it takes a little while to sink in, we’re all learning here. The key is to keep practicing and to not be afraid to make mistakes. Each mistake is an opportunity to learn and improve. Let's make sure we understand the fundamental idea behind solving equations before we get started. When we solve equations, our aim is to find the value of the unknown variable (usually represented by letters like x, y, or a) that makes the equation true. We achieve this by isolating the variable on one side of the equation. This means getting the variable by itself, with no other numbers or operations attached to it, on one side of the equals sign. To do this, we use inverse operations to “undo” anything that has been applied to the variable. For example, if we are adding a number to the variable, we subtract that number from both sides of the equation. Remember, everything we do to one side of the equation, we must do to the other to maintain the balance. Now, let’s get down to business and work on these equations. Each of them will give you different practice in applying these basic concepts and operations. Make sure to understand each step. It is very important to do practice and more problems of each type until you master the concept. Let’s do it!

a) x - 3 = 5

Here's how to solve it:

  1. Isolate x: To get x by itself, we need to get rid of the -3. We do this by adding 3 to both sides of the equation. This gives us: x - 3 + 3 = 5 + 3
  2. Simplify: This simplifies to x = 8.

So, the solution to this equation is x = 8. This means that if we substitute 8 for x in the original equation, it will hold true: 8 - 3 = 5. Now, we are going to look at other examples. The strategy remains the same: isolate the variable by using inverse operations. Each time, we will try to clearly explain how to get the correct answer. The more practice you do, the more comfortable and confident you will become. Do not rush the process; understanding each step is more important than speed. Remember that patience and persistence are key to mastering any new skill. Always remember the fundamental principle of keeping the equation balanced. Now, let’s solve another problem!

b) x - 0.5 = 1.3

Let's break it down:

  1. Isolate x: To get x by itself, we need to eliminate -0.5. Add 0.5 to both sides: x - 0.5 + 0.5 = 1.3 + 0.5
  2. Simplify: This gives us x = 1.8.

Therefore, the solution to this equation is x = 1.8. You can verify this by substituting 1.8 for x in the original equation: 1.8 - 0.5 = 1.3. Remember, the goal is always to isolate the variable. We did that by canceling out the -0.5 on the left side of the equation. Now, let’s proceed with another example. As you can see, the process is very consistent. With practice, you will solve these types of problems automatically. The process requires careful attention, especially when dealing with decimal numbers. Make sure to perform the operations correctly to avoid mistakes. The best part about math is that you can check your solutions. Now, let’s jump to the next one.

c) y - 2/3 = 1 1/6

Here's the breakdown:

  1. Convert Mixed Number: First, convert 1 1/6 to an improper fraction: 1 1/6 = 7/6.
  2. Isolate y: To get y by itself, we must add 2/3 to both sides: y - 2/3 + 2/3 = 7/6 + 2/3
  3. Find a Common Denominator: Since 2/3 is not easily added to 7/6, we must convert 2/3 to a fraction with a denominator of 6. Multiply the numerator and denominator by 2: (2/3) * (2/2) = 4/6.
  4. Add the Fractions: Now we can rewrite the equation as y = 7/6 + 4/6.
  5. Simplify: Add the numerators: y = 11/6.
  6. Convert back to mixed fraction y = 1 5/6.

So, the solution is y = 1 5/6. Remember, when dealing with fractions, finding a common denominator is crucial for performing addition and subtraction. Double check the steps of each operation to avoid mistakes. Take your time, especially with fraction arithmetic; it's easy to make a small error that changes the answer. With practice, you'll become more confident in handling fractions. Now, let’s try another equation. These types of equations are very common in different math problems, so it's a good idea to know how to resolve them. Now, we'll continue with some more examples. We'll be working with both fractions and decimals, so you'll get a well-rounded practice. Keep in mind the rules of the operations.

d) y - 2/3 = 4 1/4

Let's work this one out:

  1. Convert Mixed Number: Convert 4 1/4 to an improper fraction: 4 1/4 = 17/4.
  2. Isolate y: Add 2/3 to both sides: y - 2/3 + 2/3 = 17/4 + 2/3
  3. Find a Common Denominator: Find the least common denominator (LCD) for 4 and 3, which is 12. Convert the fractions: 17/4 = 51/12 and 2/3 = 8/12.
  4. Add the Fractions: Now we have y = 51/12 + 8/12.
  5. Simplify: Add the numerators: y = 59/12.
  6. Convert back to mixed fraction y = 4 11/12.

Therefore, y = 4 11/12. Always remember to simplify your answers when possible and convert improper fractions back to mixed numbers. Double checking the solution by substituting it into the original equation will help you spot mistakes. We're getting closer to mastering all of these problems. The more problems you solve, the more confident you'll become. Remember to take your time and review your steps. Always review your final answer. Now, we're going to proceed with some more problems. Remember that the process is always the same: isolate the variable and solve the equation. Let’s keep going!

e) a - 0.1 = 1/2

Alright, let's solve this one together:

  1. Convert Fraction to Decimal: Convert 1/2 to a decimal: 1/2 = 0.5.
  2. Isolate a: To isolate 'a,' we need to add 0.1 to both sides: a - 0.1 + 0.1 = 0.5 + 0.1.
  3. Simplify: This gives us a = 0.6.

So, the solution is a = 0.6. When dealing with fractions and decimals, you can choose to work with them separately or convert one to the other, depending on which is easier for you. Always keep in mind what you're most comfortable with. Double-check your answer by substituting the result into the original equation. Also, always review the steps to make sure you have the correct operations. This exercise is helping us to build skills and understanding. Now, we will see another case.

f) a - 1/4 = 0.65

Let's finish it:

  1. Convert Fraction to Decimal: Convert 1/4 to a decimal: 1/4 = 0.25.
  2. Isolate a: Add 0.25 to both sides: a - 0.25 + 0.25 = 0.65 + 0.25
  3. Simplify: This results in a = 0.9.

Therefore, the solution is a = 0.9. Good job, we're almost at the end! Remember the steps: isolate the variable, and always check your work by substituting the value back into the original equation. Make sure you're comfortable with both fractions and decimals. Now, let’s move on to the next set of equations!

More Equation-Solving Practice

Now, we're going to mix things up a bit with a new set of equations. Remember, the core concept remains the same: isolating the variable using inverse operations. These examples will help solidify your understanding and give you more practice with different types of numbers and operations. Let’s get started. Each problem will require us to manipulate the equation to get the variable all by itself. Let’s go!

a) a + 2.3 = 3.2

Here’s how to solve it:

  1. Isolate a: Subtract 2.3 from both sides: a + 2.3 - 2.3 = 3.2 - 2.3
  2. Simplify: This simplifies to a = 0.9.

So, the solution is a = 0.9. Easy peasy! In this case, we only need a subtraction to solve the equation. Now, let’s go to the next problem and keep practicing. Let's move on to the next one. Remember to always double-check your work, particularly when dealing with decimals. Understanding these small steps helps you grasp more complex concepts later. Now we are getting closer to the end. Just a few more problems and you'll become a champion! Now, let’s go!

b) b - 0.28 = 0.11

Here’s the breakdown:

  1. Isolate b: Add 0.28 to both sides: b - 0.28 + 0.28 = 0.11 + 0.28
  2. Simplify: This gives us b = 0.39.

Therefore, the solution is b = 0.39. Always double-check your answer and pay close attention to the decimal points. By now, you should be getting pretty comfortable with this process. With each problem, your confidence is increasing. Remember, the more you practice, the better you'll become. Let’s keep going!

c) 5.5 - c = 0.8

Let's get this one done:

  1. Isolate c: In this case, the variable is being subtracted. First, subtract 5.5 from both sides: 5.5 - c - 5.5 = 0.8 - 5.5
  2. Simplify: This simplifies to -c = -4.7
  3. Solve for c: Divide both sides by -1: -c / -1 = -4.7 / -1.
  4. Simplify: This gives us c = 4.7.

So, the solution is c = 4.7. Pay close attention when the variable is being subtracted; the process changes slightly. Remember that the objective is always to have the variable by itself. This type of equation may require a few extra steps. Make sure you don't make mistakes during the operations. Remember, the more you practice, the easier it becomes. Now we are close to the end. Let’s finish it!

d) p + 1.4 = 2 3/4

Alright, let's solve this:

  1. Convert Mixed Number: Convert 2 3/4 to a decimal: 2 3/4 = 2.75. Or convert to fraction to do operations more easily 2 3/4 = 11/4
  2. Isolate p: Subtract 1.4 from both sides: p + 1.4 - 1.4 = 2.75 - 1.4.
  3. Simplify: This gives us p = 1.35.

So, the solution is p = 1.35. Make sure you're comfortable with converting between fractions and decimals. Remember to double-check that your arithmetic is correct. Almost there! Now let's jump to the next problem. We’re doing great! Keep it up!

e) r - 5/7 = 1 6/7

Let's see this one:

  1. Convert Mixed Number: Convert 1 6/7 to an improper fraction: 1 6/7 = 13/7.
  2. Isolate r: Add 5/7 to both sides: r - 5/7 + 5/7 = 13/7 + 5/7.
  3. Simplify: This gives us r = 18/7.
  4. Convert back to mixed fraction r = 2 4/7.

So, the solution is r = 2 4/7. When adding or subtracting fractions, make sure they have a common denominator. Practice makes perfect. Keep up the good work! We're almost done. Now let's jump to the next problem. This is a great exercise to learn these operations. Let's solve the last one!

f) 2.5 - s = 1/4

Here’s how to do it:

  1. Convert Fraction to Decimal: Convert 1/4 to a decimal: 1/4 = 0.25.
  2. Isolate s: Subtract 2.5 from both sides: 2.5 - s - 2.5 = 0.25 - 2.5.
  3. Simplify: This simplifies to -s = -2.25.
  4. Solve for s: Divide both sides by -1: -s / -1 = -2.25 / -1.
  5. Simplify: This gives us s = 2.25.

So, the solution is s = 2.25. And we're done! Congratulations on making it through all the examples. Remember, the more you practice, the easier it will become. Keep up the great work. We are done! You've got this! Now you know how to solve equations.

Conclusion: You've Got This!

Congratulations, guys! You've made it through a lot of equations today. Remember, the key is practice and understanding the basics. Keep practicing, and you'll become a pro in no time. If you got stuck, don’t worry! Just go back and review the steps. The most important is that you keep trying and practicing. Keep it up! Remember the core idea: isolate the variable. You've got this. Keep practicing, and you'll be solving equations like a pro in no time! Keep practicing, keep learning, and keep asking questions. If you need any help, don’t hesitate to ask. Good luck, and have fun with math! You're well on your way to mastering equations. Now go out there and solve some problems!