Solving Equations With Mixed Numbers: A Step-by-Step Guide

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Hey everyone! Today, we're diving into solving equations that involve mixed numbers. Don't worry if you find these a bit tricky – we'll break it down step by step so you can tackle them with confidence. We've got two equations to solve, and we’ll go through each one meticulously. So, grab your pencils and let's get started!

1) Solving the First Equation: 8 5/7 - x = 4 9/14

When you're faced with an equation like 8 5/7 - x = 4 9/14, the first thing to do is isolate the variable x. In this case, x is being subtracted from a mixed number, so we need to think about how to get it by itself on one side of the equation. To do this, we can add x to both sides and subtract 4 9/14 from both sides. This keeps the equation balanced, which is super important in algebra. Remember, whatever you do to one side, you've gotta do to the other!

So, let's rewrite the equation by adding x to both sides:

8 5/7 - x + x = 4 9/14 + x

This simplifies to:

8 5/7 = 4 9/14 + x

Now, we need to subtract 4 9/14 from both sides:

8 5/7 - 4 9/14 = 4 9/14 + x - 4 9/14

This gives us:

8 5/7 - 4 9/14 = x

Okay, great! Now we have x isolated on one side. The next step is to actually perform the subtraction. But before we can subtract mixed numbers, we need to make sure they have a common denominator. The denominators we have are 7 and 14. The least common multiple of 7 and 14 is 14, so we'll convert 8 5/7 to have a denominator of 14. To do this, we multiply both the numerator and the denominator of the fractional part by 2:

5/7 * (2/2) = 10/14

So, 8 5/7 becomes 8 10/14. Now our equation looks like this:

8 10/14 - 4 9/14 = x

Now we can subtract the whole numbers and the fractions separately:

(8 - 4) + (10/14 - 9/14) = x

This simplifies to:

4 + 1/14 = x

So, x = 4 1/14. And that's our solution for the first equation! Remember, you can always double-check your answer by plugging it back into the original equation to make sure it works. This is a great way to catch any little mistakes you might have made along the way. Solving for variables with mixed numbers can be a bit tricky at first, but with practice, you’ll get the hang of it. Keep practicing, and you'll become a pro in no time!

2) Solving the Second Equation: (x + 7 5/8) - 4 13/24 = 5 1/16

Alright, let's tackle the second equation: (x + 7 5/8) - 4 13/24 = 5 1/16. This one looks a bit more complex, but don't worry, we'll break it down step by step. The key here is to follow the order of operations and isolate x carefully. First things first, we need to deal with that subtraction outside the parentheses. We can do this by adding 4 13/24 to both sides of the equation. Remember, keeping the equation balanced is crucial!

So, let's add 4 13/24 to both sides:

(x + 7 5/8) - 4 13/24 + 4 13/24 = 5 1/16 + 4 13/24

This simplifies to:

x + 7 5/8 = 5 1/16 + 4 13/24

Now, we need to add the mixed numbers on the right side of the equation. To do this, we need a common denominator for the fractions. We have denominators of 16 and 24. The least common multiple of 16 and 24 is 48, so we'll convert both fractions to have a denominator of 48.

For 1/16, we multiply the numerator and denominator by 3:

1/16 * (3/3) = 3/48

So, 5 1/16 becomes 5 3/48.

For 13/24, we multiply the numerator and denominator by 2:

13/24 * (2/2) = 26/48

So, 4 13/24 becomes 4 26/48.

Now we can rewrite the equation as:

x + 7 5/8 = 5 3/48 + 4 26/48

Let's add the mixed numbers on the right side:

5 3/48 + 4 26/48 = (5 + 4) + (3/48 + 26/48) = 9 + 29/48 = 9 29/48

So our equation now looks like this:

x + 7 5/8 = 9 29/48

To isolate x, we need to subtract 7 5/8 from both sides:

x + 7 5/8 - 7 5/8 = 9 29/48 - 7 5/8

This simplifies to:

x = 9 29/48 - 7 5/8

Again, we need a common denominator to subtract the mixed numbers. We have denominators of 48 and 8. The least common multiple is 48, so we'll convert 5/8 to have a denominator of 48. We multiply the numerator and denominator by 6:

5/8 * (6/6) = 30/48

So, 7 5/8 becomes 7 30/48. Now our equation is:

x = 9 29/48 - 7 30/48

Oops! We've got a slight problem here. We can't subtract 30/48 from 29/48 directly, so we need to borrow 1 from the whole number 9. When we borrow 1, we're actually borrowing 48/48, which we can add to the fraction:

9 29/48 = 8 + 48/48 + 29/48 = 8 77/48

Now we can rewrite the equation as:

x = 8 77/48 - 7 30/48

Subtracting the whole numbers and fractions, we get:

x = (8 - 7) + (77/48 - 30/48) = 1 + 47/48

So, x = 1 47/48. That's our solution for the second equation! Remember to check your answer by plugging it back into the original equation. This helps ensure you haven't made any errors along the way. Equations with mixed numbers can seem daunting, but with careful steps and a bit of practice, you can master them. Keep up the great work!

Key Takeaways for Solving Equations with Mixed Numbers

Alright, guys, let’s recap the main points we’ve covered today. Solving equations with mixed numbers might seem like a Herculean task at first, but once you break it down into manageable steps, it becomes much easier. Remember, the key is to stay organized and methodical. Here are some key takeaways to keep in mind:

  1. Isolate the Variable: The primary goal in solving any equation is to get the variable by itself on one side. Use inverse operations (addition, subtraction, multiplication, division) to move terms around while keeping the equation balanced. For instance, if you have x being added to a number, subtract that number from both sides. If x is being multiplied by a number, divide both sides by that number.

  2. Find a Common Denominator: When dealing with fractions (and mixed numbers), finding a common denominator is crucial for addition and subtraction. The least common multiple (LCM) of the denominators is usually the easiest choice. Convert all fractions to have this common denominator before proceeding with the operation. Remember, this involves multiplying both the numerator and the denominator by the same number to keep the fraction equivalent.

  3. Convert Mixed Numbers to Improper Fractions (If Needed): While you can add and subtract mixed numbers directly by dealing with the whole numbers and fractions separately, sometimes it’s easier to convert them to improper fractions, especially when borrowing is involved. An improper fraction is one where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and then put that result over the original denominator.

  4. Borrowing in Subtraction: When subtracting mixed numbers, you might encounter a situation where the fraction you’re subtracting is larger than the fraction you’re subtracting from. In this case, you'll need to borrow 1 from the whole number part. Remember that borrowing 1 means you're adding the denominator over itself to the fraction (e.g., borrowing 1 when the denominator is 8 means adding 8/8 to the fraction).

  5. Simplify: After performing operations, always simplify your answer. This might involve reducing fractions to their simplest form or combining like terms. Simplifying not only makes your answer look cleaner but also makes it easier to work with in future steps, if necessary.

  6. Check Your Work: This is super important! Once you've found a solution, plug it back into the original equation to make sure it works. This step can help you catch any mistakes you might have made along the way. It’s like a little insurance policy for your math!

  7. Practice, Practice, Practice: Like any skill, solving equations with mixed numbers gets easier with practice. The more problems you solve, the more comfortable and confident you’ll become. Don't get discouraged if you make mistakes – they're part of the learning process.

Solving equations with mixed numbers is a fundamental skill in algebra, and mastering it will set you up for success in more advanced math topics. So keep these tips in mind, keep practicing, and you'll be solving these equations like a pro in no time! You've got this!