Solve 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 The Equation 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.

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Introduction

Equations involving fractions can be challenging to solve, but rewriting them as proportions can make the process much easier. 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}. This equation involves three fractions, and our goal is to rewrite it as a proportion.

Rewriting the Equation as a Proportion

To rewrite the equation as a proportion, we need to find a common denominator for the fractions on the left-hand side. The common denominator is 2x2x, so we can rewrite the equation as:

1⋅x2⋅x+1⋅22⋅2=x2−7x+104x\frac{1 \cdot x}{2 \cdot x} + \frac{1 \cdot 2}{2 \cdot 2} = \frac{x^2-7x+10}{4x}

Simplifying the fractions, we get:

x2x+24=x2−7x+104x\frac{x}{2x} + \frac{2}{4} = \frac{x^2-7x+10}{4x}

Now, we can rewrite the equation as a proportion by multiplying both sides by the least common multiple (LCM) of the denominators, which is 4x4x. This gives us:

x2x⋅4x+24⋅4x=x2−7x+104x⋅4x\frac{x}{2x} \cdot 4x + \frac{2}{4} \cdot 4x = \frac{x^2-7x+10}{4x} \cdot 4x

Simplifying the fractions, we get:

2x+2x=x2−7x+102x + 2x = x^2 - 7x + 10

Finding the Equivalent Proportion

Now that we have rewritten the equation as a proportion, we can find the equivalent proportion by multiplying both sides by the reciprocal of the fraction on the left-hand side. The reciprocal of x2x\frac{x}{2x} is 2xx\frac{2x}{x}, so we can multiply both sides by this fraction to get:

2xx⋅(2x+2x)=2xx⋅(x2−7x+10)\frac{2x}{x} \cdot (2x + 2x) = \frac{2x}{x} \cdot (x^2 - 7x + 10)

Simplifying the fractions, we get:

4x+4xx=2x(x2−7x+10)x\frac{4x + 4x}{x} = \frac{2x(x^2 - 7x + 10)}{x}

Simplifying further, we get:

8xx=2x2−14x+20\frac{8x}{x} = 2x^2 - 14x + 20

Simplifying the Proportion

Now that we have found the equivalent proportion, we can simplify it by canceling out any common factors. In this case, we can cancel out the xx in the numerator and denominator to get:

8=2x2−14x+208 = 2x^2 - 14x + 20

Conclusion

In this article, we have shown 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. We have also simplified the proportion by canceling out any common factors. This process can be useful for solving equations involving fractions and can help to make the process of solving equations much easier.

Final Answer

The final answer is:

x+22x=x2−7x+104x\frac{x+2}{2x}=\frac{x^2-7x+10}{4x}

This is the equivalent proportion to the original equation.

Discussion

This problem is a great example of how to rewrite an equation as a proportion and find the equivalent proportion. It requires a good understanding of fractions and proportions, as well as the ability to simplify expressions. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

If you are struggling with fractions and proportions, there are many online resources available to help you. Some popular resources include:

  • Khan Academy: Khan Academy has a wide range of video tutorials and practice exercises on fractions and proportions.
  • Mathway: Mathway is an online math problem solver that can help you solve equations and other math problems.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve equations and other math problems.

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is often written in the form ab=cd\frac{a}{b} = \frac{c}{d}, where aa, bb, cc, and dd are numbers.

Q: How do I rewrite an equation as a proportion?

A: To rewrite an equation as a proportion, you need to find a common denominator for the fractions on the left-hand side. Then, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

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

A: The LCM of two numbers is the smallest number that both numbers can divide into evenly. For example, the LCM of 4 and 6 is 12, because both 4 and 6 can divide into 12 evenly.

Q: How do I find the equivalent proportion?

A: To find the equivalent proportion, you need to multiply both sides of the equation by the reciprocal of the fraction on the left-hand side. The reciprocal of a fraction is obtained by swapping the numerator and denominator.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of ab\frac{a}{b} is ba\frac{b}{a}.

Q: How do I simplify a proportion?

A: To simplify a proportion, you need to cancel out any common factors between the numerator and denominator. This can be done by dividing both the numerator and denominator by the common factor.

Q: What is the final answer to the original equation?

A: The final answer to the original equation is x+22x=x2−7x+104x\frac{x+2}{2x}=\frac{x^2-7x+10}{4x}.

Q: Can you provide more examples of how to rewrite an equation as a proportion?

A: Yes, here are a few more examples:

  • 13+16=x2+2x+16x\frac{1}{3} + \frac{1}{6} = \frac{x^2 + 2x + 1}{6x}
  • 25−110=x2−3x+210x\frac{2}{5} - \frac{1}{10} = \frac{x^2 - 3x + 2}{10x}
  • 34+18=x2+5x+68x\frac{3}{4} + \frac{1}{8} = \frac{x^2 + 5x + 6}{8x}

Q: Where can I find more resources on solving equations with proportions?

A: There are many online resources available to help you learn more about solving equations with proportions. Some popular resources include:

  • Khan Academy: Khan Academy has a wide range of video tutorials and practice exercises on fractions and proportions.
  • Mathway: Mathway is an online math problem solver that can help you solve equations and other math problems.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve equations and other math problems.

Conclusion

Solving equations with proportions can be a challenging but rewarding process. By following the steps outlined in this article, you can rewrite an equation as a proportion and find the equivalent proportion. Remember to find a common denominator, multiply both sides by the LCM, and simplify the proportion to get the final answer. If you have any questions or need further clarification, please don't hesitate to ask.