Solving For X In Terms Of C A Step By Step Guide

This article provides a detailed walkthrough on how to solve the equation $\frac{7}{3}(cx + \frac{1}{2}) - \frac{1}{4} = \frac{5}{2}$ for $x$ in terms of $c$. We will break down each step, ensuring a clear understanding of the algebraic manipulations involved. This problem falls under the category of linear equations and requires a solid grasp of arithmetic operations, fraction manipulation, and the distributive property. Whether you are a student tackling algebra problems or simply looking to refresh your math skills, this guide will provide you with the necessary steps and explanations to confidently solve this type of equation. Our goal is not just to provide the answer but to equip you with the knowledge to approach similar problems effectively.

Step 1: Distribute the $\frac{7}{3}$

The first crucial step in solving this equation is to distribute the $\frac{7}{3}$ across the terms inside the parenthesis. This means multiplying both $cx$ and $\frac{1}{2}$ by $\frac{7}{3}$. This process eliminates the parenthesis, simplifying the equation and making it easier to work with. The distributive property is a fundamental concept in algebra, allowing us to expand expressions and combine like terms. Failing to distribute correctly is a common mistake, so paying close attention to this step is essential for an accurate solution. Let's delve into the details of this distribution.

$\frac{7}{3}(cx + \frac{1}{2})$ becomes $\frac{7}{3} * cx + \frac{7}{3} * \frac{1}{2}$.

Performing the multiplication, we get $\frac{7cx}{3} + \frac{7}{6}$.

So, the equation now looks like this:

$\frac{7cx}{3} + \frac{7}{6} - \frac{1}{4} = \frac{5}{2}$

This distribution step is a foundational move that sets the stage for further simplification and isolation of the variable $x$. Understanding the distributive property is not just important for this problem but for a wide range of algebraic manipulations.

Step 2: Combine the Constants

After distributing, our equation contains several constant terms: $\frac{7}{6}$, $-\frac{1}{4}$, and $\frac{5}{2}$. The next logical step is to combine these constants to simplify the equation further. To do this, we need to find a common denominator for these fractions. This common denominator will allow us to add and subtract the fractions effectively. The process of combining constants is a crucial technique in solving equations, as it reduces the number of terms and makes the equation more manageable. This step requires careful attention to fraction arithmetic, including finding the least common multiple (LCM) and adjusting the numerators accordingly. Let's break down the process of finding the common denominator and combining these constants.

We need to find the least common multiple (LCM) of 6, 4, and 2. The LCM is 12.

Now, we convert each fraction to have a denominator of 12:

• $\frac{7}{6} = \frac{7 * 2}{6 * 2} = \frac{14}{12}$
• $\frac{1}{4} = \frac{1 * 3}{4 * 3} = \frac{3}{12}$
• $\frac{5}{2} = \frac{5 * 6}{2 * 6} = \frac{30}{12}$

Substituting these equivalent fractions back into the equation, we get:

$\frac{7cx}{3} + \frac{14}{12} - \frac{3}{12} = \frac{30}{12}$

Now, we can combine the fractions on the left side:

$\frac{14}{12} - \frac{3}{12} = \frac{11}{12}$

So, the equation becomes:

$\frac{7cx}{3} + \frac{11}{12} = \frac{30}{12}$

This step of combining constants is a vital simplification technique, making the equation cleaner and preparing it for the next stage of solving for x.

Step 3: Isolate the Term with x

Now that we've combined the constants, our next goal is to isolate the term containing x, which is $\frac{7cx}{3}$. To achieve this, we need to eliminate the constant term $\frac{11}{12}$ from the left side of the equation. This is accomplished by performing the inverse operation: subtracting $\frac{11}{12}$ from both sides of the equation. Maintaining balance in an equation is a core principle of algebra; whatever operation is performed on one side must also be performed on the other. This step is crucial for bringing us closer to isolating x and ultimately solving the equation. Let's walk through the subtraction process in detail.

To isolate the term with x, subtract $\frac{11}{12}$ from both sides of the equation:

$\frac{7cx}{3} + \frac{11}{12} - \frac{11}{12} = \frac{30}{12} - \frac{11}{12}$

This simplifies to:

$\frac{7cx}{3} = \frac{30 - 11}{12}$

$\frac{7cx}{3} = \frac{19}{12}$

By subtracting $\frac{11}{12}$ from both sides, we have successfully isolated the term with x on the left side, paving the way for the final steps in solving for x. This isolation technique is a fundamental algebraic maneuver, used extensively in solving various types of equations.

Step 4: Solve for x

With the term containing x isolated, we are now in the final stage of solving for x. Our equation currently looks like this: $\frac{7cx}{3} = \frac{19}{12}$. To isolate x completely, we need to get rid of the coefficients $\frac{7c}{3}$. We can do this by multiplying both sides of the equation by the reciprocal of $\frac{7c}{3}$, which is $\frac{3}{7c}$. Remember, multiplying by the reciprocal is equivalent to dividing. This step involves careful manipulation of fractions and algebraic terms, ensuring we apply the operation correctly to both sides to maintain the equation's balance. Let's break down the process of isolating x in detail.

To solve for x, multiply both sides of the equation by $\frac{3}{7c}$:

$\frac{7cx}{3} * \frac{3}{7c} = \frac{19}{12} * \frac{3}{7c}$

On the left side, $\frac{7c}{3}$ and $\frac{3}{7c}$ cancel each other out, leaving us with just x:

$x = \frac{19}{12} * \frac{3}{7c}$

Now, we multiply the fractions on the right side:

$x = \frac{19 * 3}{12 * 7c}$

$x = \frac{57}{84c}$

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

$x = \frac{57 ÷ 3}{84c ÷ 3}$

$x = \frac{19}{28c}$

Upon reviewing the answer choices provided in the prompt, it appears there was an error in the simplification or in the provided options. The correct solution, obtained through careful algebraic manipulation, is $x = \frac{19}{28c}$. The options listed do not match this result. It is important to double-check both the problem setup and the solution steps to ensure accuracy in such cases. The process we followed, distributing, combining constants, isolating the x term, and finally solving for x, is a standard approach to solving linear equations. Understanding each step and the underlying algebraic principles is key to success in solving these types of problems.

Conclusion

In summary, we have walked through the step-by-step process of solving the equation $\frac{7}{3}(cx + \frac{1}{2}) - \frac{1}{4} = \frac{5}{2}$ for x in terms of c. The key steps involved distributing the fraction, combining constant terms, isolating the term with x, and finally, solving for x by multiplying by the reciprocal. While our calculations led us to the solution $x = \frac{19}{28c}$, it's crucial to note that this result does not match any of the provided answer choices, suggesting a potential error in the original options or a need for careful re-evaluation of the problem setup. This exercise highlights the importance of meticulous algebraic manipulation and verification of results. Solving equations for a specific variable is a fundamental skill in algebra and mathematics as a whole. The systematic approach we've demonstrated here can be applied to a wide variety of problems, reinforcing the importance of understanding the underlying principles rather than just memorizing steps. Keep practicing, and you'll become more confident in your ability to tackle algebraic equations!