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

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 linear equation involving fractions and a constant term. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

The Given Equation

The given equation is:

$$\frac{x}{3}+\frac{x}{4}+15=x$$

Our goal is to solve for the value of $x$.

Step 1: Eliminate the Fractions

To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators, which are 3 and 4. The LCM of 3 and 4 is 12.

We can multiply both sides of the equation by 12 to eliminate the fractions:

$$12\left(\frac{x}{3}+\frac{x}{4}+15\right)=12x$$

This simplifies to:

$$4x+3x+180=12x$$

Step 2: Combine Like Terms

We can combine the like terms on the left-hand side of the equation:

$$7x+180=12x$$

Step 3: Isolate the Variable

To isolate the variable $x$, we need to get all the terms with $x$ on one side of the equation. We can do this by subtracting $7x$ from both sides:

$$180=5x$$

Step 4: Solve for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by 5:

$$x=\frac{180}{5}$$

$$x=36$$

Conclusion

In this article, we solved a linear equation involving fractions and a constant term. We broke down the solution into four steps: eliminating the fractions, combining like terms, isolating the variable, and solving for $x$. By following these steps, we were able to find the value of $x$, which is 36.

Answer

The correct answer is:

• A. 18: Incorrect
• B. 24: Incorrect
• C. 36: Correct
• D. 48: Incorrect
• E. 60: Incorrect

Final Thoughts

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, and $x$ is the variable.

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

A: The steps to solve a linear equation are:

1. Eliminate any fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
2. Combine like terms on the left-hand side of the equation.
3. Isolate the variable by getting all the terms with the variable on one side of the equation.
4. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: How do I eliminate fractions in a linear equation?

A: To eliminate fractions in a linear equation, you need to find the least common multiple (LCM) of the denominators and multiply both sides of the equation by the LCM.

For example, if the equation is:

$$\frac{x}{3}+\frac{x}{4}+15=x$$

The LCM of 3 and 4 is 12. Multiplying both sides of the equation by 12 gives:

$$12\left(\frac{x}{3}+\frac{x}{4}+15\right)=12x$$

This simplifies to:

$$4x+3x+180=12x$$

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, the equation:

$$x+2=5$$

is a linear equation, while the equation:

$$x^2+4x+4=0$$

is a quadratic equation.

Q: Can I use algebraic manipulations to solve a linear equation?

A: Yes, you can use algebraic manipulations to solve a linear equation. Some common algebraic manipulations include:

• Adding or subtracting the same value to both sides of the equation
• Multiplying or dividing both sides of the equation by the same value
• Using the distributive property to expand expressions

For example, if the equation is:

$$2x+5=11$$

You can subtract 5 from both sides to get:

$$2x=6$$

Then, you can divide both sides by 2 to get:

$$x=3$$

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

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

• Forgetting to eliminate fractions
• Not combining like terms
• Not isolating the variable
• Making errors when multiplying or dividing both sides of the equation

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate.

Conclusion

Solving linear equations is an essential skill for students to master. By following the steps outlined in this article and being aware of common mistakes to avoid, you can develop a deeper understanding of how to solve linear equations and apply this skill to a wide range of mathematical problems.