Select The Correct Answer.Solve The Equation For X X X In Terms Of C C C . 2 3 ( C X + 1 2 ) − 1 4 = 5 2 \frac{2}{3}\left(c X+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2} 3 2 ​ ( C X + 2 1 ​ ) − 4 1 ​ = 2 5 ​ A. X = 9 4 C X=\frac{9}{4 C} X = 4 C 9 ​ B. X = 27 8 C X=\frac{27}{8 C} X = 8 C 27 ​ C. $x=\frac{29}{8

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Understanding the Problem

When solving equations, it's essential to isolate the variable of interest. In this case, we're given an equation involving the variable $x$ and a constant $c$. Our goal is to express $x$ in terms of $c$. To do this, we'll need to manipulate the equation using algebraic operations.

The Given Equation

The equation we're working with is:

$$\frac{2}{3}\left(c x+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2}$$

Step 1: Simplify the Equation

To simplify the equation, we can start by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 12.

12 * (2/3 * (cx + 1/2) - 1/4) = 12 * (5/2)

Step 2: Distribute the Multiplication

Now, we can distribute the multiplication to simplify the equation further.

(8 * cx + 6) - 3 = 30

Step 3: Combine Like Terms

Next, we can combine like terms on the left-hand side of the equation.

8cx + 3 = 30

Step 4: Isolate the Term with $x$

To isolate the term with $x$, we can subtract 3 from both sides of the equation.

8cx = 27

Step 5: Solve for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by 8c.

x = 27 / (8c)

Conclusion

After simplifying the equation and isolating the term with $x$, we found that the solution is:

$$x = \frac{27}{8c}$$

This is the correct answer.

Discussion

The equation we solved is a linear equation involving a variable $x$ and a constant $c$. We used algebraic operations to isolate the term with $x$ and express it in terms of $c$. The solution we found is a rational expression, which is a common type of solution in algebra.

Comparison with Other Options

Let's compare our solution with the other options given:

• Option A: $x = \frac{9}{4c}$
• Option B: $x = \frac{27}{8c}$
• Option C: $x = \frac{29}{8c}$

Our solution matches option B, which is the correct answer.

Final Thoughts

Solving equations is an essential skill in mathematics, and it's crucial to understand the steps involved in isolating variables and expressing them in terms of other variables. In this case, we used algebraic operations to simplify the equation and find the solution. By following these steps, you can solve equations and express variables in terms of other variables.

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Understanding the Problem

When solving equations, it's essential to isolate the variable of interest. In this case, we're given an equation involving the variable $x$ and a constant $c$. Our goal is to express $x$ in terms of $c$. To do this, we'll need to manipulate the equation using algebraic operations.

Q&A

Q: What is the given equation?

A: The given equation is:

$$\frac{2}{3}\left(c x+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2}$$

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify it by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 12.

Q: How do we distribute the multiplication in the equation?

A: We distribute the multiplication by multiplying each term inside the parentheses by 12.

Q: What is the result of combining like terms on the left-hand side of the equation?

A: After combining like terms, the equation becomes:

$$8cx + 3 = 30$$

Q: How do we isolate the term with $x$?

A: To isolate the term with $x$, we subtract 3 from both sides of the equation.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to divide both sides of the equation by 8c to solve for $x$.

Q: What is the solution to the equation?

A: The solution to the equation is:

$$x = \frac{27}{8c}$$

Q: How does the solution compare to the other options?

A: Our solution matches option B, which is the correct answer.

Q: What is the importance of solving equations?

A: Solving equations is an essential skill in mathematics, and it's crucial to understand the steps involved in isolating variables and expressing them in terms of other variables.

Discussion

The equation we solved is a linear equation involving a variable $x$ and a constant $c$. We used algebraic operations to isolate the term with $x$ and express it in terms of $c$. The solution we found is a rational expression, which is a common type of solution in algebra.

Tips and Tricks

• When solving equations, it's essential to simplify the equation by getting rid of fractions.
• Use algebraic operations to isolate the term with the variable of interest.
• Combine like terms to simplify the equation.
• Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Conclusion

Solving equations is an essential skill in mathematics, and it's crucial to understand the steps involved in isolating variables and expressing them in terms of other variables. By following these steps, you can solve equations and express variables in terms of other variables.