What Is Another Way To Write The Equation 7 8 X + 3 4 = − 6 \frac{7}{8} X+\frac{3}{4}=-6 8 7 ​ X + 4 3 ​ = − 6 ?A. 7 ( X 8 ) + 3 4 = − 6 7\left(\frac{x}{8}\right)+\frac{3}{4}=-6 7 ( 8 X ​ ) + 4 3 ​ = − 6 B. 7 + 3 8 + 4 X = − 6 \frac{7+3}{8+4} X=-6 8 + 4 7 + 3 ​ X = − 6 C. 7 8 + 3 4 X = − 6 \frac{7}{8}+\frac{3}{4} X=-6 8 7 ​ + 4 3 ​ X = − 6 D. $\frac{7}{8}

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Introduction

When dealing with linear equations, it's essential to understand various ways to represent them. This knowledge can help in simplifying complex equations and making them more manageable. In this article, we'll explore an alternative way to write the equation 78x+34=−6\frac{7}{8} x+\frac{3}{4}=-6.

The Original Equation

The given equation is 78x+34=−6\frac{7}{8} x+\frac{3}{4}=-6. This equation represents a linear relationship between the variable xx and the constant term −6-6. The coefficients of xx and the constant term are fractions, which can make the equation appear more complicated.

Alternative Representations

Let's examine the options provided to rewrite the equation:

Option A: Distributive Property

Option A is 7(x8)+34=−67\left(\frac{x}{8}\right)+\frac{3}{4}=-6. This representation uses the distributive property to rewrite the fraction 78\frac{7}{8} as 7(x8)7\left(\frac{x}{8}\right). This is a valid way to represent the equation, but it may not be the most straightforward or simplified form.

Option B: Adding Fractions

Option B is 7+38+4x=−6\frac{7+3}{8+4} x=-6. This representation involves adding the numerators and denominators of the fractions separately. However, this approach is incorrect because it doesn't take into account the coefficients of xx.

Option C: Incorrect Order

Option C is 78+34x=−6\frac{7}{8}+\frac{3}{4} x=-6. This representation is incorrect because it adds the fractions 78\frac{7}{8} and 34\frac{3}{4}, which is not the correct operation to perform when simplifying the equation.

Option D: Correct Representation

Option D is 78x+34=−61\frac{7}{8} x+\frac{3}{4}=\frac{-6}{1}. This representation is the correct way to rewrite the equation. It involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 8. This results in a simplified equation with integer coefficients.

Simplifying the Equation

To simplify the equation, we can multiply both sides by the LCM of the denominators, which is 8. This gives us:

78x+34=−61\frac{7}{8} x+\frac{3}{4}=\frac{-6}{1}

Multiplying both sides by 8, we get:

7x+6=−487x+6=-48

Subtracting 6 from both sides, we get:

7x=−547x=-54

Dividing both sides by 7, we get:

x=−547x=-\frac{54}{7}

Conclusion

In conclusion, the correct way to rewrite the equation 78x+34=−6\frac{7}{8} x+\frac{3}{4}=-6 is 78x+34=−61\frac{7}{8} x+\frac{3}{4}=\frac{-6}{1}. This representation involves multiplying both sides of the equation by the LCM of the denominators, which is 8. This results in a simplified equation with integer coefficients. By simplifying the equation, we can solve for the variable xx.

Final Answer

The final answer is: −547\boxed{-\frac{54}{7}}

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Introduction

In our previous article, we explored alternative ways to represent the equation 78x+34=−6\frac{7}{8} x+\frac{3}{4}=-6. We also simplified the equation and solved for the variable xx. In this article, we'll answer some frequently asked questions related to simplifying linear equations.

Q&A

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 8 and 4 is 8, because 8 is a multiple of both 8 and 4.

Q: How do I find the LCM of two fractions?

A: To find the LCM of two fractions, you need to find the LCM of the denominators. For example, if you have the fractions 18\frac{1}{8} and 14\frac{1}{4}, the LCM of the denominators is 8.

Q: Can I simplify an equation by multiplying both sides by a number?

A: Yes, you can simplify an equation by multiplying both sides by a number. However, you need to make sure that the number you multiply by is not zero, and that it is not a fraction with a denominator of zero.

Q: How do I simplify an equation with fractions?

A: To simplify an equation with fractions, you need to find the LCM of the denominators and multiply both sides of the equation by that number. This will eliminate the fractions and make the equation easier to solve.

Q: Can I add or subtract fractions in an equation?

A: No, you cannot add or subtract fractions in an equation unless they have the same denominator. If the fractions have different denominators, you need to find the LCM of the denominators and multiply both sides of the equation by that number before adding or subtracting the fractions.

Q: How do I solve an equation with fractions?

A: To solve an equation with fractions, you need to follow these steps:

  1. Find the LCM of the denominators and multiply both sides of the equation by that number.
  2. Simplify the equation by eliminating the fractions.
  3. Add or subtract the fractions as needed.
  4. Solve for the variable.

Example

Let's say we have the equation 23x+14=−2\frac{2}{3} x+\frac{1}{4}=-2. To solve this equation, we need to follow the steps above.

  1. Find the LCM of the denominators: The LCM of 3 and 4 is 12.
  2. Multiply both sides of the equation by 12: 23x+14=−2⇒8x+3=−24\frac{2}{3} x+\frac{1}{4}=-2 \Rightarrow 8x+3=-24
  3. Simplify the equation: 8x=−278x=-27
  4. Solve for the variable: x=−278x=-\frac{27}{8}

Conclusion

In conclusion, simplifying linear equations with fractions requires finding the LCM of the denominators and multiplying both sides of the equation by that number. This will eliminate the fractions and make the equation easier to solve. By following the steps above, you can solve equations with fractions and find the value of the variable.

Final Answer

The final answer is: −278\boxed{-\frac{27}{8}}