If $\frac{x}{3}+\frac{x}{4}+15=x$, Then $x =$A. 18 B. 24 C. 36 D. 48 E. 60

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which involves fractions and a constant term. We will use the given equation x3+x4+15=x\frac{x}{3}+\frac{x}{4}+15=x to demonstrate the step-by-step process of solving linear equations.

Understanding the Equation

The given equation is x3+x4+15=x\frac{x}{3}+\frac{x}{4}+15=x. To solve this equation, we need to isolate the variable xx on one side of the equation. The equation involves fractions, so we will need to find a common denominator to combine the fractions.

Step 1: Find a Common Denominator

To find a common denominator, we need to identify the denominators of the fractions in the equation. In this case, the denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12. Therefore, we will multiply the first fraction by 44\frac{4}{4} and the second fraction by 33\frac{3}{3} to get a common denominator of 12.

\frac{x}{3} \cdot \frac{4}{4} = \frac{4x}{12}
\frac{x}{4} \cdot \frac{3}{3} = \frac{3x}{12}

Step 2: Combine the Fractions

Now that we have a common denominator, we can combine the fractions by adding them together.

\frac{4x}{12} + \frac{3x}{12} = \frac{7x}{12}

Step 3: Subtract 15 from Both Sides

Next, we need to subtract 15 from both sides of the equation to isolate the term with the variable xx.

\frac{7x}{12} - 15 = x

Step 4: Multiply Both Sides by 12

To eliminate the fraction, we need to multiply both sides of the equation by 12.

12 \cdot \frac{7x}{12} - 12 \cdot 15 = 12x
7x - 180 = 12x

Step 5: Subtract 7x from Both Sides

Now, we need to subtract 7x from both sides of the equation to isolate the constant term.

7x - 7x - 180 = 12x - 7x
-180 = 5x

Step 6: Divide Both Sides by 5

Finally, we need to divide both sides of the equation by 5 to solve for xx.

\frac{-180}{5} = \frac{5x}{5}
-36 = x

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation involving fractions and a constant term. By following these steps, we have successfully isolated the variable xx and solved for its value. The final answer is x=−36x = -36.

Discussion

This equation is a classic example of a linear equation, and solving it requires a clear understanding of fractions and algebraic manipulation. The steps outlined in this article provide a clear and concise guide for students to follow when solving similar equations. By practicing these steps, students can develop their problem-solving skills and become more confident in their ability to solve linear equations.

Answer Options

A. 18 B. 24 C. 36 D. 48 E. 60

Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation involving fractions and a constant term. In this article, we will provide a Q&A guide to help students better understand the concepts and techniques involved in solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  3. Check the solution by plugging it back into the original equation.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. For example, 2x and 3x are like terms, but 2x and 3y are not.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, x + 2 = 3 is a linear equation, while x^2 + 2x + 1 = 0 is a quadratic equation.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to find a common denominator and then combine the fractions. Once you have combined the fractions, you can isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: The order of operations when solving a linear equation is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug it back into the original equation and see if it is true. If the solution satisfies the original equation, then it is a valid solution.

Conclusion

In this article, we have provided a Q&A guide to help students better understand the concepts and techniques involved in solving linear equations. By following the steps outlined in this article, students can develop their problem-solving skills and become more confident in their ability to solve linear equations.

Common Mistakes to Avoid

  • Not simplifying the equation before solving it.
  • Not isolating the variable correctly.
  • Not checking the solution.
  • Not following the order of operations.

Tips for Solving Linear Equations

  • Read the equation carefully and understand what it is asking for.
  • Simplify the equation before solving it.
  • Isolate the variable correctly.
  • Check the solution.
  • Follow the order of operations.

Practice Problems

  1. Solve the equation: 2x + 5 = 11
  2. Solve the equation: x - 3 = 7
  3. Solve the equation: 4x + 2 = 14

Answer Key

  1. x = 3
  2. x = 10
  3. x = 3