Solving For X: A Step-by-Step Guide

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Hey guys! Today, we're diving into a common math problem: solving for x in an equation. Specifically, we're tackling the equation 6.75 + (3/8)x = 13 1/4. Don't worry, it might look a little intimidating at first, but we'll break it down step-by-step so it's super easy to understand. We’ll walk through each part, ensuring you grasp not just the how, but also the why behind each step. Mastering this kind of problem is crucial because it forms the foundation for more complex algebraic equations. Stick with me, and you’ll be solving these like a pro in no time! So, let’s put on our math hats and get started, ensuring you feel confident and clear about each step we take.

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation is telling us. We have:

6.75 + (3/8)x = 13 1/4

This equation basically says that if we take 6.75 and add it to (3/8) multiplied by some unknown number x, the result will be 13 1/4. Our goal is to figure out what that unknown number x is. Remember, equations are like a balancing act; whatever we do to one side, we have to do to the other to keep things balanced. This principle is key to solving for x! The left side of the equation includes a decimal, a fraction multiplied by a variable, and a constant, while the right side is a mixed number. Converting these into a common format will be our initial focus to streamline the solving process. So, let’s delve into how we can manipulate these numbers to make our equation-solving journey smoother and more understandable.

Step 1: Convert Mixed Numbers and Decimals to Fractions

To make things easier to work with, let's convert the decimal and mixed number into fractions. This will help us avoid dealing with different types of numbers and keep everything consistent. First, let’s tackle the decimal 6.75. We can convert this to a fraction by recognizing that 0.75 is the same as 3/4. So, 6.75 can be written as 6 3/4.

Next, we need to convert this mixed number into an improper fraction. To do this, we multiply the whole number (6) by the denominator (4) and then add the numerator (3). This gives us (6 * 4) + 3 = 27. We then place this over the original denominator, giving us 27/4. So, 6.75 is equal to 27/4.

Now, let's convert the mixed number 13 1/4 into an improper fraction. We do the same thing: multiply the whole number (13) by the denominator (4) and add the numerator (1). This gives us (13 * 4) + 1 = 53. Place this over the original denominator to get 53/4.

Now our equation looks like this:

27/4 + (3/8)x = 53/4

Converting these numbers into fractions is a crucial step as it unifies the numerical format, making it simpler to perform algebraic manipulations. By doing this, we’ve set the stage for the next steps in solving for x, ensuring that our calculations will be straightforward and accurate. This unified approach is a common strategy in algebra for handling equations with diverse numerical representations.

Step 2: Isolate the Term with x

Our next goal is to isolate the term with x on one side of the equation. This means we want to get (3/8)x by itself. To do this, we need to get rid of the 27/4 that's being added on the left side. Remember, we need to keep the equation balanced, so whatever we do to one side, we must do to the other. We can eliminate 27/4 by subtracting it from both sides of the equation:

(27/4 + (3/8)x) - 27/4 = (53/4) - 27/4

On the left side, 27/4 and -27/4 cancel each other out, leaving us with just (3/8)x. On the right side, we subtract the fractions: 53/4 - 27/4. Since they have the same denominator, we can simply subtract the numerators: 53 - 27 = 26. So, we have 26/4. Now our equation looks like this:

(3/8)x = 26/4

Isolating the term with x is a fundamental step in solving algebraic equations. By performing the same operation on both sides, we maintain the equation's balance and progressively simplify it. This step brings us closer to unveiling the value of x by separating the variable from other terms, setting up the final calculation to find x.

Step 3: Solve for x

Now we have the equation (3/8)x = 26/4. To solve for x, we need to get rid of the (3/8) that's multiplying it. We can do this by multiplying both sides of the equation by the reciprocal of 3/8, which is 8/3. This is because multiplying a fraction by its reciprocal equals 1, effectively isolating x. So, let's multiply both sides by 8/3:

(8/3) * (3/8)x = (26/4) * (8/3)

On the left side, (8/3) * (3/8) equals 1, so we're left with just x. On the right side, we multiply the fractions: (26/4) * (8/3). Before multiplying, we can simplify by canceling common factors. Notice that 4 and 8 have a common factor of 4. Divide both by 4 to get 1 and 2, respectively. So, our multiplication becomes (26/1) * (2/3). Now we multiply the numerators and the denominators: 26 * 2 = 52, and 1 * 3 = 3. This gives us 52/3.

So, our equation now looks like this:

x = 52/3

To convert this improper fraction to a mixed number, we divide 52 by 3. 3 goes into 52 seventeen times (17 * 3 = 51) with a remainder of 1. So, 52/3 is equal to 17 1/3. Therefore, our final answer is:

x = 17 1/3

Solving for x by multiplying by the reciprocal is a critical technique in algebra. This step effectively undoes the multiplication affecting x, allowing us to isolate the variable and determine its value. Converting the improper fraction to a mixed number helps in understanding the magnitude of x in a more practical context. This final step provides a clear and concise solution to the original equation.

Step 4: Verification (Optional but Recommended)

To make sure we got the correct answer, it's always a good idea to plug our value of x back into the original equation and see if it holds true. Our original equation was:

6.75 + (3/8)x = 13 1/4

We found that x = 17 1/3. Let's substitute this value into the equation:

6. 75 + (3/8) * (17 1/3) = 13 1/4

First, we need to convert 17 1/3 to an improper fraction. 17 * 3 + 1 = 52, so 17 1/3 = 52/3. Now our equation looks like this:

6.75 + (3/8) * (52/3) = 13 1/4

Next, let's multiply (3/8) by (52/3). We can simplify by canceling the common factor of 3: (1/8) * 52. Now we have 52/8. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4. 52 ÷ 4 = 13, and 8 ÷ 4 = 2. So, 52/8 simplifies to 13/2.

Now our equation looks like this:

6.75 + 13/2 = 13 1/4

Let's convert 6.75 to a fraction, which we already know is 27/4, and convert 13/2 to have a denominator of 4. To do this, we multiply the numerator and denominator by 2: (13 * 2) / (2 * 2) = 26/4. Now our equation is:

27/4 + 26/4 = 13 1/4

Add the fractions on the left side: 27/4 + 26/4 = 53/4. Now our equation is:

53/4 = 13 1/4

Finally, convert 13 1/4 to an improper fraction: 13 * 4 + 1 = 53, so 13 1/4 = 53/4. Now we have:

53/4 = 53/4

Since both sides of the equation are equal, our solution for x is correct!

Verification is a super important step in problem-solving. It confirms the accuracy of our solution by ensuring it satisfies the original equation. This process not only validates our answer but also deepens our understanding of the equation and the steps involved. By substituting our calculated value back into the equation, we ensure that our solution maintains the balance and integrity of the mathematical relationship.

Conclusion

So, we've successfully solved for x in the equation 6.75 + (3/8)x = 13 1/4, and we found that x = 17 1/3. We did this by converting decimals and mixed numbers to fractions, isolating the term with x, and then multiplying by the reciprocal. Remember, the key to solving algebraic equations is to keep the equation balanced and perform the same operations on both sides. And don't forget to check your work by plugging your answer back into the original equation! You got this! If you practice these steps, you'll be solving for x like a math whiz in no time. Keep up the great work, and remember, every problem you solve makes you stronger in math!