Solving Equations Step-by-Step Guide With Examples
Hey guys! Today, we're diving into the exciting world of solving equations. It might sound intimidating, but trust me, it's like piecing together a puzzle. We'll break down the process step-by-step, making it super easy to understand. We'll tackle some equations with fractions, which can sometimes look scary, but don't worry, we've got this! So, grab your pencils and notebooks, and let's get started on this math adventure together. Remember, practice makes perfect, so the more equations you solve, the better you'll become at it. Think of each equation as a mini-challenge, and with each challenge you overcome, you're leveling up your math skills. We'll use a mix of clear explanations, helpful tips, and real examples to guide you along the way. So, buckle up and get ready to become equation-solving pros!
1) Solving 2 2/11 x - 5/16 = 1 3/4
Let's dive into solving our first equation: 2 2/11 x - 5/16 = 1 3/4. This might look a bit complex with those mixed numbers and fractions, but don't worry, we'll break it down into manageable steps. Our main goal here is to isolate the variable 'x' on one side of the equation. To do this, we'll perform a series of operations, making sure to keep the equation balanced. First things first, let's convert those mixed numbers into improper fractions. This will make our calculations a whole lot easier. Remember, a mixed number is a combination of a whole number and a fraction, and an improper fraction has a numerator that is greater than or equal to its denominator. Converting to improper fractions allows us to work with a single fraction instead of two separate parts. So, let's get those conversions done and then we'll move on to the next step in solving for 'x'. Once we have our equation in terms of improper fractions, we can start thinking about how to isolate 'x'. This usually involves adding or subtracting constants from both sides of the equation, and then multiplying or dividing by the coefficient of 'x'. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain balance. This is a fundamental principle in solving equations, and it ensures that we're not changing the solution.
Converting Mixed Numbers to Improper Fractions
Okay, let's tackle those mixed numbers. We have 2 2/11 and 1 3/4. To convert 2 2/11 to an improper fraction, we multiply the whole number (2) by the denominator (11) and add the numerator (2). This gives us (2 * 11) + 2 = 24. Then, we keep the same denominator, so 2 2/11 becomes 24/11. Easy peasy, right? Now, let's do the same for 1 3/4. We multiply the whole number (1) by the denominator (4) and add the numerator (3). This gives us (1 * 4) + 3 = 7. We keep the same denominator, so 1 3/4 becomes 7/4. See? It's just a couple of simple steps. Now that we've converted our mixed numbers to improper fractions, our equation looks a bit different, but it's actually in a form that's much easier to work with. We're one step closer to isolating 'x' and finding our solution. Remember, these conversions are a crucial first step in solving equations with mixed numbers, so it's important to get comfortable with the process. The more you practice, the quicker and more confident you'll become. Now that we have our improper fractions, we can move on to the next stage of the equation-solving adventure!
Rewriting the Equation
With our mixed numbers converted, our equation now looks like this: 24/11 x - 5/16 = 7/4. See how much cleaner that looks? Now, our goal is still to isolate 'x', but we need to get rid of that pesky 5/16 first. To do that, we'll add 5/16 to both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced. This is like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. Adding 5/16 to both sides will cancel out the - 5/16 on the left side, leaving us with just the term containing 'x'. This is a key step in isolating the variable, and it's a technique that you'll use again and again in solving equations. Once we've added 5/16 to both sides, we'll need to simplify the right side by finding a common denominator and adding the fractions. This might seem like a bit of extra work, but it's essential for getting our equation into its simplest form. So, let's add those fractions and see what our equation looks like then! We're making progress towards our solution, one step at a time.
Adding 5/16 to Both Sides
Okay, let's add 5/16 to both sides of our equation: 24/11 x - 5/16 + 5/16 = 7/4 + 5/16. As you can see, the - 5/16 and + 5/16 on the left side cancel each other out, leaving us with 24/11 x on the left. Now, we need to add 7/4 and 5/16 on the right side. But, we can't add fractions unless they have the same denominator! So, we need to find a common denominator for 4 and 16. The least common multiple of 4 and 16 is 16, so we'll convert 7/4 to an equivalent fraction with a denominator of 16. To do this, we multiply both the numerator and denominator of 7/4 by 4: (7 * 4) / (4 * 4) = 28/16. Now we can add the fractions: 28/16 + 5/16 = 33/16. So, our equation now looks like this: 24/11 x = 33/16. We're getting closer and closer to isolating 'x'. We've eliminated one term on the left side and combined the fractions on the right side. Now, we just need to get rid of that 24/11 that's multiplying 'x'. Are you ready for the next step? Let's keep going!
Multiplying by the Reciprocal
Now we have 24/11 x = 33/16. To isolate 'x', we need to get rid of the fraction 24/11 that's multiplying it. The easiest way to do this is to multiply both sides of the equation by the reciprocal of 24/11, which is 11/24. Remember, the reciprocal of a fraction is just flipping it upside down. Multiplying a fraction by its reciprocal always equals 1, so 24/11 multiplied by 11/24 will just give us 1, leaving 'x' all by itself on the left side. This is the magic of using reciprocals! On the right side, we'll need to multiply 33/16 by 11/24. This might seem a bit daunting, but we can simplify things by looking for common factors in the numerators and denominators before we multiply. This can save us a lot of time and effort in the long run. Once we've multiplied both sides by the reciprocal and simplified, we'll have our solution for 'x'. So, let's get those fractions multiplied and see what we get! We're almost there – just a few more steps and we'll have solved our first equation.
So, let's multiply both sides by 11/24: (11/24) * (24/11) x = (33/16) * (11/24). On the left side, (11/24) * (24/11) cancels out to 1, leaving us with just x. On the right side, we have (33/16) * (11/24). Before we multiply, let's see if we can simplify. We can divide 33 and 24 by 3, giving us 11 and 8, respectively. So, we now have (11/16) * (11/8). Multiplying the numerators gives us 121, and multiplying the denominators gives us 128. So, x = 121/128. Woohoo! We solved it! That might have seemed like a lot of steps, but we broke it down and tackled each part one at a time. Remember, the key is to stay organized, keep your equations balanced, and don't be afraid of fractions! We've conquered our first equation, and we're ready to move on to the next challenge. Let's keep practicing and building our equation-solving skills!
2) Solving 4 2/9 x + 3 5/14 = 6 11/21
Alright, let's jump into our second equation: 4 2/9 x + 3 5/14 = 6 11/21. Just like before, we're going to break this down into easy-to-manage steps. We've got those mixed numbers again, so our first mission is to convert them into improper fractions. This is a crucial step in making the equation easier to work with. Once we've got our improper fractions, we'll start thinking about isolating 'x'. Remember, that's the name of the game! We'll need to get rid of any terms that are being added or subtracted from the 'x' term, and then we'll need to deal with any coefficient that's multiplying 'x'. We'll use the same techniques we used in the first equation, making sure to keep the equation balanced at every step. Think of it as a delicate balancing act – we need to perform the same operations on both sides to maintain equilibrium. So, let's get those mixed numbers converted and see what our equation looks like in its new form. We're on our way to finding the value of 'x', one step at a time!
Converting Mixed Numbers to Improper Fractions
Time to transform those mixed numbers! We've got 4 2/9, 3 5/14, and 6 11/21. Let's start with 4 2/9. We multiply the whole number (4) by the denominator (9) and add the numerator (2): (4 * 9) + 2 = 38. We keep the same denominator, so 4 2/9 becomes 38/9. Next up is 3 5/14. We multiply the whole number (3) by the denominator (14) and add the numerator (5): (3 * 14) + 5 = 47. We keep the same denominator, so 3 5/14 becomes 47/14. Last but not least, we have 6 11/21. We multiply the whole number (6) by the denominator (21) and add the numerator (11): (6 * 21) + 11 = 137. We keep the same denominator, so 6 11/21 becomes 137/21. Now our equation is looking much more fraction-friendly! We've successfully converted all those mixed numbers into improper fractions, and we're ready to move on to the next stage of solving for 'x'. Remember, this conversion process is a fundamental skill in working with equations that involve mixed numbers, so it's well worth mastering. Now that we've got our improper fractions, we can start thinking about how to isolate 'x'. Let's see what our equation looks like now!
Rewriting the Equation
With our conversions complete, our equation now reads: 38/9 x + 47/14 = 137/21. That's a lot of fractions, but don't let it intimidate you! Our goal remains the same: isolate 'x'. To do this, we need to get rid of that 47/14 that's being added to the 38/9 x term. So, what do we do? We subtract 47/14 from both sides of the equation, of course! This will cancel out the + 47/14 on the left side, leaving us with just the term containing 'x'. Remember, the golden rule of equation solving is to do the same thing to both sides. This keeps the equation balanced and ensures that we're not changing the solution. Once we've subtracted 47/14 from both sides, we'll need to simplify the right side by finding a common denominator and subtracting the fractions. This might take a little bit of work, but it's a necessary step in getting our equation into its simplest form. So, let's subtract those fractions and see what we've got. We're making steady progress towards finding the value of 'x'!
Subtracting 47/14 from Both Sides
Let's subtract 47/14 from both sides: 38/9 x + 47/14 - 47/14 = 137/21 - 47/14. The + 47/14 and - 47/14 on the left side cancel each other out, leaving us with 38/9 x. Now we need to subtract 47/14 from 137/21 on the right side. To do this, we need to find a common denominator for 21 and 14. The least common multiple of 21 and 14 is 42, so we'll convert both fractions to equivalent fractions with a denominator of 42. To convert 137/21 to a fraction with a denominator of 42, we multiply both the numerator and denominator by 2: (137 * 2) / (21 * 2) = 274/42. To convert 47/14 to a fraction with a denominator of 42, we multiply both the numerator and denominator by 3: (47 * 3) / (14 * 3) = 141/42. Now we can subtract the fractions: 274/42 - 141/42 = 133/42. So, our equation now looks like this: 38/9 x = 133/42. We've successfully isolated the term with 'x' on the left side and simplified the right side. We're one step closer to finding the value of 'x'. Are you ready for the next move? Let's keep going!
Multiplying by the Reciprocal
We're at 38/9 x = 133/42. To get 'x' all by itself, we need to get rid of the 38/9 that's multiplying it. Just like before, we'll multiply both sides of the equation by the reciprocal of 38/9, which is 9/38. Multiplying a fraction by its reciprocal always results in 1, so (9/38) * (38/9) will cancel out, leaving us with just 'x' on the left side. On the right side, we'll need to multiply 133/42 by 9/38. This might look a bit intimidating, but remember, we can simplify things by looking for common factors before we multiply. This can make our calculations much easier. Once we've multiplied both sides by the reciprocal and simplified, we'll have our solution for 'x'. So, let's get those fractions multiplied and see what we get! We're almost there – just a few more steps and we'll have conquered our second equation.
Let's multiply both sides by 9/38: (9/38) * (38/9) x = (133/42) * (9/38). On the left side, (9/38) * (38/9) cancels out to 1, leaving us with x. On the right side, we have (133/42) * (9/38). Let's simplify before we multiply. We can divide 42 and 9 by 3, giving us 14 and 3, respectively. We can also divide 133 and 38 by 19, giving us 7 and 2, respectively. So, we now have (7/14) * (3/2). We can further simplify 7/14 to 1/2. So, our equation is now (1/2) * (3/2). Multiplying the numerators gives us 3, and multiplying the denominators gives us 4. So, x = 3/4. Awesome! We've solved another equation! We converted mixed numbers, subtracted fractions, and multiplied by reciprocals. We tackled each step with confidence and found our solution. Remember, practice is key, so keep working on these skills and you'll become an equation-solving master!
3) Solving 11/18 - 14/27 x = 5/12
Okay, guys, let's move on to our third equation: 11/18 - 14/27 x = 5/12. This one looks a little different, but the same principles apply. Our ultimate goal is still to isolate 'x', and we'll do that by carefully manipulating the equation. Notice that we don't have any mixed numbers this time, which is a nice little break! We already have fractions, so we can jump right into the process of isolating 'x'. The first thing we'll want to do is get rid of that 11/18 that's being added to the - 14/27 x term. How do we do that? You guessed it – we'll subtract 11/18 from both sides of the equation. This will keep the equation balanced and move us closer to our goal. Once we've subtracted 11/18 from both sides, we'll need to simplify the right side by finding a common denominator and subtracting the fractions. This is a familiar step by now, and we know we can handle it. So, let's get started by subtracting those fractions and see what our equation looks like then! We're on our way to solving for 'x', one step at a time.
Subtracting 11/18 from Both Sides
Alright, let's subtract 11/18 from both sides of the equation: 11/18 - 14/27 x - 11/18 = 5/12 - 11/18. On the left side, the 11/18 and - 11/18 cancel each other out, leaving us with - 14/27 x. Now, we need to subtract 11/18 from 5/12 on the right side. To do this, we need to find a common denominator for 12 and 18. The least common multiple of 12 and 18 is 36, so we'll convert both fractions to equivalent fractions with a denominator of 36. To convert 5/12 to a fraction with a denominator of 36, we multiply both the numerator and denominator by 3: (5 * 3) / (12 * 3) = 15/36. To convert 11/18 to a fraction with a denominator of 36, we multiply both the numerator and denominator by 2: (11 * 2) / (18 * 2) = 22/36. Now we can subtract the fractions: 15/36 - 22/36 = -7/36. So, our equation now looks like this: - 14/27 x = -7/36. We've successfully isolated the term with 'x' on the left side and simplified the right side. We're getting closer and closer to finding the value of 'x'. Let's keep the momentum going!
Multiplying by the Reciprocal
We've reached - 14/27 x = -7/36. To isolate 'x', we need to get rid of the - 14/27 that's multiplying it. As we've done before, we'll multiply both sides of the equation by the reciprocal of - 14/27, which is - 27/14. Remember, the reciprocal is just flipping the fraction, and we keep the negative sign. Multiplying a fraction by its reciprocal gives us 1, so (- 27/14) * (- 14/27) will cancel out, leaving us with just 'x' on the left side. On the right side, we'll need to multiply - 7/36 by - 27/14. Notice that we're multiplying two negative numbers, which will result in a positive number. This is an important detail to keep in mind! We can also simplify things by looking for common factors before we multiply. This will make our calculations much easier and reduce the chances of making a mistake. Once we've multiplied both sides by the reciprocal and simplified, we'll have our solution for 'x'. So, let's get those fractions multiplied and see what the answer is! We're almost there – just a final push and we'll have solved our third equation.
Let's multiply both sides by - 27/14: (- 27/14) * (- 14/27) x = (- 7/36) * (- 27/14). On the left side, (- 27/14) * (- 14/27) cancels out to 1, leaving us with just x. On the right side, we have (- 7/36) * (- 27/14). Since we're multiplying two negative numbers, the result will be positive. Let's simplify before we multiply. We can divide 7 and 14 by 7, giving us 1 and 2, respectively. We can also divide 27 and 36 by 9, giving us 3 and 4, respectively. So, we now have (1/4) * (3/2). Multiplying the numerators gives us 3, and multiplying the denominators gives us 8. So, x = 3/8. Boom! We did it! We solved our third equation. We subtracted fractions, multiplied by reciprocals, and remembered to handle those negative signs carefully. You guys are becoming equation-solving pros! Remember, the more you practice, the more confident and skilled you'll become. So, keep challenging yourselves with new equations, and you'll be amazed at how much you can achieve.
Conclusion
Great job, everyone! We've successfully tackled three different equations, and along the way, we've reinforced some key concepts in algebra. We converted mixed numbers to improper fractions, subtracted fractions, and multiplied by reciprocals. We also emphasized the importance of keeping equations balanced by performing the same operations on both sides. These are fundamental skills that will serve you well in your math journey. Remember, solving equations is like piecing together a puzzle – each step brings you closer to the final solution. Don't be afraid to break down complex problems into smaller, more manageable steps. And most importantly, practice, practice, practice! The more equations you solve, the more comfortable and confident you'll become. So, keep up the great work, and keep exploring the exciting world of algebra!
