Solve The Equation 1 2 + 1 2 X = X 2 − 7 X + 10 4 X \frac{1}{2} + \frac{1}{2x} = \frac{x^2 - 7x + 10}{4x} 2 1 ​ + 2 X 1 ​ = 4 X X 2 − 7 X + 10 ​ By Rewriting It As A Proportion.Which Proportion Is Equivalent To The Original Equation?A. X + 2 2 X = X 2 − 7 X + 10 4 X \frac{x+2}{2x} = \frac{x^2 - 7x + 10}{4x} 2 X X + 2 ​ = 4 X X 2 − 7 X + 10 ​ B. $\frac{x-1}{2x}

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Introduction

In mathematics, equations can be solved using various methods, including rewriting them as proportions. This approach can simplify the equation and make it easier to solve. In this article, we will explore how to rewrite the equation 12+12x=x2−7x+104x\frac{1}{2} + \frac{1}{2x} = \frac{x^2 - 7x + 10}{4x} as a proportion and find the equivalent proportion.

Understanding the Original Equation

The original equation is 12+12x=x2−7x+104x\frac{1}{2} + \frac{1}{2x} = \frac{x^2 - 7x + 10}{4x}. To rewrite this equation as a proportion, we need to understand the concept of proportions. A proportion is a statement that two ratios are equal. In this case, we want to find a proportion that is equivalent to the original equation.

Rewriting the Equation as a Proportion

To rewrite the equation as a proportion, we can start by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is 4x4x. Multiplying both sides by 4x4x gives us:

2x+2=x2−7x+102x + 2 = x^2 - 7x + 10

Now, we can rewrite this equation as a proportion by dividing both sides by 2x2x. This gives us:

2x+22x=x2−7x+102x\frac{2x + 2}{2x} = \frac{x^2 - 7x + 10}{2x}

However, we can simplify this proportion further by factoring the numerator and denominator. Factoring the numerator gives us:

2(x+1)2x=x2−7x+102x\frac{2(x + 1)}{2x} = \frac{x^2 - 7x + 10}{2x}

Now, we can simplify the proportion by canceling out the common factor of 2x2x in the numerator and denominator. This gives us:

x+1x=x2−7x+102x\frac{x + 1}{x} = \frac{x^2 - 7x + 10}{2x}

However, this is not the correct proportion. We need to find a proportion that is equivalent to the original equation.

Finding the Equivalent Proportion

To find the equivalent proportion, we can start by multiplying both sides of the original equation by 4x4x. This gives us:

2x2+2=x2−7x+102x^2 + 2 = x^2 - 7x + 10

Now, we can rewrite this equation as a proportion by dividing both sides by 2x2x. This gives us:

2x2+22x=x2−7x+102x\frac{2x^2 + 2}{2x} = \frac{x^2 - 7x + 10}{2x}

However, we can simplify this proportion further by factoring the numerator and denominator. Factoring the numerator gives us:

2(x2+1)2x=x2−7x+102x\frac{2(x^2 + 1)}{2x} = \frac{x^2 - 7x + 10}{2x}

Now, we can simplify the proportion by canceling out the common factor of 2x2x in the numerator and denominator. This gives us:

x2+1x=x2−7x+102x\frac{x^2 + 1}{x} = \frac{x^2 - 7x + 10}{2x}

However, this is not the correct proportion. We need to a proportion that is equivalent to the original equation.

The Correct Proportion

After re-examining the original equation, we can see that the correct proportion is:

x−12x=x2−7x+104x\frac{x-1}{2x} = \frac{x^2 - 7x + 10}{4x}

This proportion is equivalent to the original equation, and it can be verified by multiplying both sides by 4x4x and simplifying.

Conclusion

Introduction

In our previous article, we explored how to rewrite the equation 12+12x=x2−7x+104x\frac{1}{2} + \frac{1}{2x} = \frac{x^2 - 7x + 10}{4x} as a proportion. We found that the correct proportion is x−12x=x2−7x+104x\frac{x-1}{2x} = \frac{x^2 - 7x + 10}{4x}. In this article, we will answer some common questions related to solving the equation using a proportional approach.

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

A: The LCM of the denominators is 4x4x. This is the smallest multiple that both denominators can divide into evenly.

Q: How do I multiply both sides of the equation by the LCM?

A: To multiply both sides of the equation by the LCM, you need to multiply each term in the equation by the LCM. In this case, you would multiply each term by 4x4x.

Q: How do I simplify the proportion after multiplying both sides by the LCM?

A: After multiplying both sides by the LCM, you can simplify the proportion by canceling out any common factors in the numerator and denominator. In this case, you can cancel out the common factor of 2x2x in the numerator and denominator.

Q: What is the correct proportion?

A: The correct proportion is x−12x=x2−7x+104x\frac{x-1}{2x} = \frac{x^2 - 7x + 10}{4x}. This proportion is equivalent to the original equation, and it can be used to solve the equation.

Q: How do I verify the correct proportion?

A: To verify the correct proportion, you can multiply both sides of the equation by 4x4x and simplify. If the resulting equation is true, then the proportion is correct.

Q: What are some common mistakes to avoid when rewriting the equation as a proportion?

A: Some common mistakes to avoid when rewriting the equation as a proportion include:

  • Not multiplying both sides of the equation by the LCM
  • Not simplifying the proportion after multiplying both sides by the LCM
  • Not canceling out common factors in the numerator and denominator
  • Not verifying the correct proportion

Q: What are some tips for solving the equation using a proportional approach?

A: Some tips for solving the equation using a proportional approach include:

  • Make sure to multiply both sides of the equation by the LCM
  • Simplify the proportion after multiplying both sides by the LCM
  • Cancel out common factors in the numerator and denominator
  • Verify the correct proportion
  • Be careful when canceling out common factors to avoid making mistakes

Conclusion

In this article, we answered some common questions related to solving the equation using a proportional approach. We hope that this article has been helpful in clarifying any confusion and providing additional guidance on how to solve the equation.