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.1. 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

In mathematics, solving equations is a fundamental concept that involves manipulating algebraic expressions to isolate the variable. One effective method for solving equations is by 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 solve for the variable.

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. We can rewrite the fractions as follows:

12=1⋅x2⋅x=x2x\frac{1}{2}=\frac{1\cdot x}{2\cdot x}=\frac{x}{2x}

12x=1⋅22⋅2x=24x\frac{1}{2x}=\frac{1\cdot 2}{2\cdot 2x}=\frac{2}{4x}

Now, we can rewrite the equation as a proportion:

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

Simplifying the Proportion

To simplify the proportion, we can combine the fractions on the left-hand side:

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

Equating the Numerators

Since the denominators are the same, we can equate the numerators:

x+2=x2−7x+10x+2=x^2-7x+10

Rearranging the Equation

To solve for the variable, we need to rearrange the equation:

x2−8x+8=0x^2-8x+8=0

Factoring the Quadratic Equation

The quadratic equation can be factored as follows:

(x−4)2=0(x-4)^2=0

Solving for the Variable

To solve for the variable, we can take the square root of both sides:

x−4=0x-4=0

x=4x=4

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 solve for the variable. By simplifying the proportion and equating the numerators, we were able to rearrange the equation and factor the quadratic expression. Finally, we solved for the variable by taking the square root of both sides. This approach demonstrates the effectiveness of rewriting equations as proportions in solving algebraic expressions.

Answer

The proportion equivalent to the original equation is:

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

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 and solve for the variable. In this article, we will address some common questions and provide additional insights into solving equations using this approach.

Q: What is the purpose of rewriting an equation as a proportion?

A: Rewriting an equation as a proportion can simplify the equation and make it easier to solve. By finding a common denominator and combining fractions, we can eliminate complex algebraic expressions and isolate the variable.

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

A: You can rewrite an equation as a proportion when the fractions on the left-hand side have a common denominator. This is often the case when the fractions have a common factor or when the denominators are related in some way.

Q: What if the equation has multiple fractions with different denominators?

A: In this case, you can find the least common multiple (LCM) of the denominators and rewrite each fraction with the LCM as the denominator. This will allow you to combine the fractions and rewrite the equation as a proportion.

Q: Can I use this approach to solve equations with variables in the denominator?

A: Yes, you can use this approach to solve equations with variables in the denominator. However, you will need to be careful when simplifying the equation and avoid dividing by zero.

Q: How do I know if the equation has been simplified correctly?

A: To ensure that the equation has been simplified correctly, you can check that the fractions on the left-hand side have a common denominator and that the numerators are equal. You can also use algebraic manipulations to verify that the equation is equivalent to the original equation.

Q: Can I use this approach to solve equations with multiple variables?

A: Yes, you can use this approach to solve equations with multiple variables. However, you will need to be careful when simplifying the equation and avoid introducing extraneous solutions.

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

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

  • Not finding the common denominator
  • Not combining fractions correctly
  • Introducing extraneous solutions
  • Dividing by zero

Q: How can I practice solving equations using this approach?

A: You can practice solving equations using this approach by working through examples and exercises. You can also use online resources and math software to help you visualize the equations and simplify the algebraic manipulations.

Conclusion

In this article, we have addressed some common questions and provided additional insights into solving equations using the proportional approach. By rewriting the equation as a proportion, we can simplify the equation and make it easier to solve. We hope that this article has been helpful in clarifying the process and providing a deeper understanding of this approach.

Additional Resources ----------------* Mathway: An online math software that can help you visualize equations and simplify algebraic manipulations.

  • Khan Academy: A free online resource that provides video lessons and exercises on algebra and other math topics.
  • Wolfram Alpha: A powerful online calculator that can help you solve equations and visualize mathematical concepts.