Select The Correct Answer.Solve The Equation For $x$ In Terms Of $c$. 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 = 29 8 C X=\frac{29}{8 C} X = 8 C 29 ​ B. X = 29 18 C X=\frac{29}{18 C} X = 18 C 29 ​ C. $x=\frac{9}{4

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Introduction

Equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific equation involving variables and constants. We will break down the solution step by step, making it easy to understand and follow.

The Equation

The given equation is:

23(cx+12)−14=52\frac{2}{3}\left(c x+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2}

Our goal is to solve for xx in terms of cc.

Step 1: Simplify the Equation

To simplify the equation, we will start by isolating the term involving xx. We can do this by adding 14\frac{1}{4} to both sides of the equation:

23(cx+12)=52+14\frac{2}{3}\left(c x+\frac{1}{2}\right)=\frac{5}{2}+\frac{1}{4}

Next, we will simplify the right-hand side of the equation by finding a common denominator:

23(cx+12)=104+14\frac{2}{3}\left(c x+\frac{1}{2}\right)=\frac{10}{4}+\frac{1}{4}

23(cx+12)=114\frac{2}{3}\left(c x+\frac{1}{2}\right)=\frac{11}{4}

Step 2: Distribute the Coefficient

Now that we have isolated the term involving xx, we can distribute the coefficient 23\frac{2}{3} to the terms inside the parentheses:

cx+12=32⋅114c x+\frac{1}{2}=\frac{3}{2}\cdot\frac{11}{4}

cx+12=338c x+\frac{1}{2}=\frac{33}{8}

Step 3: Isolate the Term Involving xx

To isolate the term involving xx, we will subtract 12\frac{1}{2} from both sides of the equation:

cx=338−12c x=\frac{33}{8}-\frac{1}{2}

Next, we will simplify the right-hand side of the equation by finding a common denominator:

cx=338−48c x=\frac{33}{8}-\frac{4}{8}

cx=298c x=\frac{29}{8}

Step 4: Solve for xx

Finally, we can solve for xx by dividing both sides of the equation by cc:

x=298cx=\frac{29}{8 c}

Conclusion

In this article, we solved the equation 23(cx+12)−14=52\frac{2}{3}\left(c x+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2} for xx in terms of cc. We broke down the solution into four steps, making it easy to understand and follow. The final solution is x=298cx=\frac{29}{8 c}.

Answer

The correct answer is:

  • A. x=298cx=\frac{29}{8 c}

Discussion

This equation is a linear equation in one variable, and we solved it using basic algebraic manipulations. The equation involves a constant cc and a variable xx, and we solved for xx in terms of cc. This type of equation is commonly encountered in mathematics and science, and solving it is an essential skill for students and professionals alike.

Tips and Variations

  • To solve this equation, we used basic algebraic manipulations, such as adding and subtracting terms, and distributing coefficients.
  • We also used the concept of equivalent ratios to simplify the equation.
  • This type of equation can be solved using various methods, including substitution and elimination.
  • In some cases, the equation may involve more complex operations, such as multiplication and division of fractions.

Practice Problems

  • Solve the equation 34(x+13)−12=56\frac{3}{4}\left(x+\frac{1}{3}\right)-\frac{1}{2}=\frac{5}{6} for xx in terms of a constant cc.
  • Solve the equation 25(x+14)−310=710\frac{2}{5}\left(x+\frac{1}{4}\right)-\frac{3}{10}=\frac{7}{10} for xx in terms of a constant cc.

References

Glossary

  • Algebraic Manipulation: The process of using mathematical operations to simplify or transform an equation.
  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Equivalent Ratio: A ratio that is equal to another ratio, often used to simplify equations.
    Solving Equations: A Q&A Guide =====================================

Introduction

In our previous article, we solved the equation 23(cx+12)−14=52\frac{2}{3}\left(c x+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2} for xx in terms of cc. In this article, we will provide a Q&A guide to help you understand the solution and apply it to similar problems.

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

A: The first step in solving the equation is to simplify the equation by isolating the term involving xx. We can do this by adding 14\frac{1}{4} to both sides of the equation.

Q: How do I simplify the equation?

A: To simplify the equation, we can use basic algebraic manipulations, such as adding and subtracting terms, and distributing coefficients. We can also use the concept of equivalent ratios to simplify the equation.

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

A: The next step in solving the equation is to distribute the coefficient 23\frac{2}{3} to the terms inside the parentheses. This will allow us to isolate the term involving xx.

Q: How do I isolate the term involving xx?

A: To isolate the term involving xx, we can subtract 12\frac{1}{2} from both sides of the equation. This will allow us to solve for xx in terms of cc.

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

A: The final step in solving the equation is to solve for xx by dividing both sides of the equation by cc. This will give us the solution to the equation.

Q: What are some common mistakes to avoid when solving equations?

A: Some common mistakes to avoid when solving equations include:

  • Not simplifying the equation before solving for xx
  • Not isolating the term involving xx before solving for xx
  • Not using the correct operations to solve for xx

Q: How can I apply this solution to similar problems?

A: To apply this solution to similar problems, you can follow these steps:

  • Simplify the equation by isolating the term involving xx
  • Distribute the coefficient to the terms inside the parentheses
  • Isolate the term involving xx by subtracting or adding terms
  • Solve for xx by dividing both sides of the equation by the coefficient

Q: What are some tips for solving equations?

A: Some tips for solving equations include:

  • Simplify the equation before solving for xx
  • Use the correct operations to solve for xx
  • Check your work by plugging the solution back into the original equation

Q: How can I practice solving equations?

A: To practice solving equations, you can try the following:

  • Solve equations with different coefficients and constants
  • Solve equations with different variables and operations
  • Try solving equations with multiple steps and operations

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the equation 23(cx+12)−14=52\frac{2}{3}\left(c x+\frac{1}{2}\right)-\frac{1}{4}=\frac{5}{2} and apply it to similar problems. We also provided tips and practice problems to help you improve your skills in solving equations.

Practice Problems

  • Solve the equation 34(x+13)−12=56\frac{3}{4}\left(x+\frac{1}{3}\right)-\frac{1}{2}=\frac{5}{6} for xx in terms of a constant cc.
  • Solve the equation 25(x+14)−310=710\frac{2}{5}\left(x+\frac{1}{4}\right)-\frac{3}{10}=\frac{7}{10} for xx in terms of a constant cc.

References

Glossary

  • Algebraic Manipulation: The process of using mathematical operations to simplify or transform an equation.
  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Equivalent Ratio: A ratio that is equal to another ratio, often used to simplify equations.