Solving For X In Terms Of C A Step-by-Step Guide

Jul 25, 2025 by ADMIN 49 views

Hey guys! Today, we're diving into a math problem that involves solving for x in terms of c. It might seem a bit tricky at first, but don't worry, we'll break it down step by step so you can ace it. The equation we're tackling is:

(2/3)(cx + 1/2) - 1/4 = 5/2

Our mission is to isolate x on one side of the equation. Let's get started!

Understanding the Equation

Before we jump into the solution, let's take a moment to understand what this equation is all about. We have a linear equation where x is the variable we want to find, and c is another variable that will be part of our solution. This means our final answer will express x in terms of c. The equation involves fractions, so we'll need to be comfortable with fraction arithmetic to solve it effectively. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Keeping this in mind will help us navigate the steps smoothly.

Think of it like this: we're trying to unwrap a present. The x is hidden inside, and we need to carefully undo each layer (operation) to get to it. Each step we take will bring us closer to isolating x and expressing it in terms of c. So, let’s put on our math hats and get unwrapping!

To begin, the left side of the equation has a term multiplied by 2/3, which includes both cx and 1/2 within the parentheses. We also have a -1/4 subtracted from this term. On the right side, we have 5/2. Our goal is to manipulate the equation using algebraic operations to get x by itself. This involves distributing the 2/3, combining like terms, and using inverse operations to isolate x. The final form of our solution will look like x = some expression involving c. This kind of problem is common in algebra and is a fundamental skill for more advanced math topics. So, mastering this now will set you up for success later on. Are you ready to dive into the solution? Let's do it!

Step 1: Distribute the 2/3

The first thing we need to do is distribute the 2/3 across the terms inside the parentheses. This means we'll multiply 2/3 by both cx and 1/2. Let's break it down:

(2/3) * (cx) = (2/3)cx

(2/3) * (1/2) = 2/6 = 1/3

So, after distributing, our equation looks like this:

(2/3)cx + 1/3 - 1/4 = 5/2

Distributing the 2/3 is a crucial first step because it helps us get rid of the parentheses and makes it easier to combine like terms later on. Think of it as expanding the equation so we can see all the individual pieces clearly. This is a common technique in algebra, and you'll use it in many different types of problems. It's like opening up the inner workings of a machine so you can see how each part contributes to the whole. Now that we've distributed, we're one step closer to isolating x. Remember, the goal is to get x all by itself on one side of the equation, and each step we take should bring us closer to that goal. So, let's keep moving forward and see what the next step entails. We've handled the parentheses; now we need to deal with the fractions. Are you ready to tackle the next challenge? Let's go!

This step highlights the importance of understanding the distributive property, which is a fundamental concept in algebra. By correctly applying this property, we simplify the equation and make it easier to work with. So, always remember to distribute carefully and double-check your work to ensure accuracy. A small mistake in this step can throw off the entire solution, so it's worth taking the time to get it right. And remember, practice makes perfect! The more you work with the distributive property, the more comfortable you'll become with it. So, keep practicing, and you'll be solving these kinds of equations like a pro in no time!

Step 2: Combine the Constants

Now, let's focus on the constants on the left side of the equation: 1/3 and -1/4. To combine them, we need to find a common denominator. The least common multiple of 3 and 4 is 12, so we'll convert both fractions to have a denominator of 12.

1/3 = (1/3) * (4/4) = 4/12

-1/4 = (-1/4) * (3/3) = -3/12

Now we can combine them:

4/12 - 3/12 = 1/12

Our equation now looks like this:

(2/3)cx + 1/12 = 5/2

Combining constants is a key step in simplifying equations. It's like tidying up the equation so we can focus on the main task: isolating x. By finding a common denominator and adding or subtracting the fractions, we reduce the number of terms and make the equation more manageable. This is a fundamental skill in algebra, and you'll use it frequently when solving equations. Think of it as organizing your workspace before starting a project. By clearing away the clutter, you can focus on the task at hand and avoid making mistakes. So, always take the time to combine like terms and simplify your equations as much as possible. It will make your life much easier in the long run! And remember, attention to detail is crucial in math. A small error in combining constants can lead to a wrong answer, so double-check your work and make sure everything adds up correctly.

This step emphasizes the importance of working with fractions confidently. If you're not comfortable with fraction arithmetic, it's worth taking some time to review the basics. Understanding how to find common denominators and add, subtract, multiply, and divide fractions is essential for solving algebraic equations. So, don't be afraid to brush up on your fraction skills! And remember, practice is key. The more you work with fractions, the more comfortable you'll become with them. So, keep practicing, and you'll be a fraction master in no time!

Step 3: Isolate the Term with x

Next, we want to isolate the term with x, which is (2/3)cx. To do this, we'll subtract 1/12 from both sides of the equation:

(2/3)cx + 1/12 - 1/12 = 5/2 - 1/12

(2/3)cx = 5/2 - 1/12

Now we need to subtract the fractions on the right side. Again, we need a common denominator. The least common multiple of 2 and 12 is 12, so we'll convert 5/2 to have a denominator of 12:

5/2 = (5/2) * (6/6) = 30/12

Now we can subtract:

30/12 - 1/12 = 29/12

Our equation now looks like this:

(2/3)cx = 29/12

Isolating the term with x is a crucial step because it brings us closer to our goal of solving for x. By subtracting the constant term from both sides of the equation, we're essentially clearing the way for the final steps. This is a common technique in algebra, and you'll use it frequently when solving equations. Think of it as peeling back the layers of an onion to get to the core. Each step we take removes a layer and brings us closer to the center. So, always remember to isolate the term with x as one of the key steps in your problem-solving process. And remember, maintaining balance is essential in algebra. Whatever operation you perform on one side of the equation, you must perform on the other side to keep the equation balanced. This is a fundamental principle that underlies all algebraic manipulations. So, always be mindful of maintaining balance, and you'll be well on your way to solving equations successfully!

This step highlights the importance of understanding inverse operations. Subtraction is the inverse operation of addition, so by subtracting 1/12 from both sides of the equation, we effectively undo the addition of 1/12 on the left side. This is a powerful technique in algebra, and you'll use it frequently when solving equations. So, make sure you understand the concept of inverse operations and how to apply them effectively. And remember, practice is key. The more you work with inverse operations, the more comfortable you'll become with them. So, keep practicing, and you'll be solving equations like a pro in no time!

Step 4: Solve for x

Finally, to solve for x, we need to get rid of the (2/3)c that's multiplying it. We can do this by dividing both sides of the equation by (2/3)c. Dividing by a fraction is the same as multiplying by its reciprocal, so we'll multiply both sides by 3/(2c):

(2/3)cx * (3/(2c)) = (29/12) * (3/(2c))

On the left side, the (2/3) and (3/2) cancel out, and the c's cancel out, leaving us with just x:

x = (29/12) * (3/(2c))

Now we can simplify the right side:

x = (29 * 3) / (12 * 2c)

x = 87 / (24c)

We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

x = (87 / 3) / (24c / 3)

x = 29 / (8c)

So, our final answer is:

x = 29 / (8c)

Solving for x is the ultimate goal of this process. By dividing both sides of the equation by the coefficient of x, we isolate x and express it in terms of the other variables. This is a fundamental skill in algebra, and you'll use it frequently when solving equations. Think of it as unlocking the final piece of the puzzle. Each step we've taken has led us to this point, and now we can see the complete solution. So, always remember to solve for x as the final step in your problem-solving process. And remember, simplifying your answer is just as important as finding it. Always look for opportunities to reduce fractions and express your answer in the simplest possible form. This will make your answer easier to understand and use in future calculations. So, take the extra time to simplify your answer, and you'll be sure to impress your teachers and your friends!

This step highlights the importance of understanding reciprocals and how they're used in division. Dividing by a fraction is the same as multiplying by its reciprocal, so by multiplying both sides of the equation by 3/(2c), we effectively undo the multiplication by (2/3)c on the left side. This is a powerful technique in algebra, and you'll use it frequently when solving equations. So, make sure you understand the concept of reciprocals and how to apply them effectively. And remember, practice is key. The more you work with reciprocals, the more comfortable you'll become with them. So, keep practicing, and you'll be solving equations like a pro in no time!

Step 5: Select the Correct Answer

Looking back at the original question, we see the following answer choices:

A. x = 9 / (4c) B. x = 29 / (18c) C. x = 27 / (8c) D. x = 29 / (8c)

Our solution, x = 29 / (8c), matches answer choice D. So, the correct answer is D.

Selecting the correct answer is the final step in the problem-solving process. By carefully comparing your solution to the answer choices, you can ensure that you've arrived at the correct answer. This is a crucial step, as it allows you to double-check your work and avoid making careless mistakes. Think of it as the final inspection before shipping a product. You want to make sure everything is perfect before it leaves your hands. So, always take the time to select the correct answer carefully, and you'll be sure to get the problem right! And remember, showing your work is just as important as finding the answer. By writing out each step of your solution, you can help your teacher or professor understand your thought process and give you partial credit even if you make a mistake. So, always show your work, and you'll be sure to maximize your score!

This step highlights the importance of accuracy and attention to detail. Math problems often have multiple steps, and a small mistake in any step can lead to a wrong answer. So, it's essential to be careful and double-check your work at each step. And remember, practice is key. The more you practice solving math problems, the better you'll become at identifying and avoiding mistakes. So, keep practicing, and you'll be solving problems like a pro in no time!

Conclusion

There you have it! We successfully solved for x in terms of c. Remember, the key is to break down the problem into smaller, manageable steps. Distribute, combine like terms, isolate the variable, and simplify. With practice, you'll become a pro at solving these types of equations.

Solving for x in terms of c is a fundamental skill in algebra. By mastering this skill, you'll be well-prepared for more advanced math topics. Remember, the key is to break down the problem into smaller, manageable steps and to work carefully and systematically. And remember, practice is key. The more you practice solving these types of equations, the more comfortable you'll become with them. So, keep practicing, and you'll be solving equations like a pro in no time! And remember, don't be afraid to ask for help if you get stuck. Math can be challenging, but it's also incredibly rewarding. So, keep learning, keep growing, and keep pushing yourself to achieve your full potential!

This conclusion emphasizes the importance of perseverance and a positive attitude. Math can be challenging, but it's also incredibly rewarding. By approaching math with a growth mindset and a willingness to learn from your mistakes, you can achieve your goals and reach your full potential. So, always remember to stay positive, keep practicing, and never give up on your dreams!

Key Takeaways:

Distribute to eliminate parentheses.
Combine like terms to simplify the equation.
Isolate the variable by using inverse operations.
Simplify your final answer.
Double-check your work to ensure accuracy.

Keep practicing, and you'll master these skills in no time! You got this!