Rewrite $2+\frac{1}{b-2}=\frac{3b}{b+2}$ As A Proportion. Which Of These Proportions Is Equivalent To The Original Equation?A. $\frac{3}{b-2}=\frac{3b}{b+2}$B. \$\frac{2b+3}{b-2}=\frac{3b}{b+2}$[/tex\]C.

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Understanding the Original Equation

The given equation is $2+\frac{1}{b-2}=\frac{3b}{b+2}.$ This equation involves fractions and a variable, making it a complex expression. Our goal is to rewrite this equation as a proportion, which will help us simplify and solve it more easily.

Rewriting the Equation as a Proportion

To rewrite the equation as a proportion, we need to isolate the fractions on one side of the equation. We can start by subtracting 2 from both sides of the equation:

1b−2=3bb+2−2\frac{1}{b-2}=\frac{3b}{b+2}-2

Now, we can simplify the right-hand side of the equation by finding a common denominator:

1b−2=3bb+2−2(b+2)b+2\frac{1}{b-2}=\frac{3b}{b+2}-\frac{2(b+2)}{b+2}

Combining the fractions on the right-hand side, we get:

1b−2=3b−2b−4b+2\frac{1}{b-2}=\frac{3b-2b-4}{b+2}

Simplifying the numerator, we get:

1b−2=b−4b+2\frac{1}{b-2}=\frac{b-4}{b+2}

Now, we can rewrite the equation as a proportion by multiplying both sides by the reciprocal of the denominator:

1b−2⋅(b−2)=b−4b+2⋅(b−2)\frac{1}{b-2} \cdot (b-2) = \frac{b-4}{b+2} \cdot (b-2)

This simplifies to:

1=(b−4)(b−2)b+21 = \frac{(b-4)(b-2)}{b+2}

Evaluating the Options

Now that we have rewritten the equation as a proportion, we can evaluate the given options:

A. $\frac{3}{b-2}=\frac{3b}{b+2}$

B. $\frac{2b+3}{b-2}=\frac{3b}{b+2}$

C. $\frac{b-4}{b+2}=\frac{3}{b-2}$

Let's analyze each option:

  • Option A: This option is not equivalent to the original equation. The numerator on the left-hand side is 3, while the numerator on the right-hand side is 3b. These are not equal, so option A is not correct.
  • Option B: This option is not equivalent to the original equation. The numerator on the left-hand side is 2b+3, while the numerator on the right-hand side is 3b. These are not equal, so option B is not correct.
  • Option C: This option is equivalent to the original equation. The numerator on the left-hand side is b-4, while the numerator on the right-hand side is 3. These are equal, so option C is correct.

Conclusion

Understanding the Original Equation

The given equation is $2+\frac{1}{b-2}=\frac{3b}{b+2}.$ This equation involves fractions and a variable, making it a complex expression. Our goal is to rewrite this equation as a proportion, which will help us simplify and solve it more easily.

Rewriting the Equation as a Proportion

To rewrite the equation as a proportion, we need to isolate the fractions on one side of the equation. We can start by subtracting 2 from both sides of the equation:

1b−2=3bb+2−2\frac{1}{b-2}=\frac{3b}{b+2}-2

Now, we can simplify the right-hand side of the equation by finding a common denominator:

1b−2=3bb+2−2(b+2)b+2\frac{1}{b-2}=\frac{3b}{b+2}-\frac{2(b+2)}{b+2}

Combining the fractions on the right-hand side, we get:

1b−2=3b−2b−4b+2\frac{1}{b-2}=\frac{3b-2b-4}{b+2}

Simplifying the numerator, we get:

1b−2=b−4b+2\frac{1}{b-2}=\frac{b-4}{b+2}

Now, we can rewrite the equation as a proportion by multiplying both sides by the reciprocal of the denominator:

1b−2⋅(b−2)=b−4b+2⋅(b−2)\frac{1}{b-2} \cdot (b-2) = \frac{b-4}{b+2} \cdot (b-2)

This simplifies to:

1=(b−4)(b−2)b+21 = \frac{(b-4)(b-2)}{b+2}

Evaluating the Options

Now that we have rewritten the equation as a proportion, we can evaluate the given options:

A. $\frac{3}{b-2}=\frac{3b}{b+2}$

B. $\frac{2b+3}{b-2}=\frac{3b}{b+2}$

C. $\frac{b-4}{b+2}=\frac{3}{b-2}$

Let's analyze each option:

  • Option A: This option is not equivalent to the original equation. The numerator on the left-hand side is 3, while the numerator on the right-hand side is 3b. These are not equal, so option A is not correct.
  • Option B: This option is not equivalent to the original equation. The numerator on the left-hand side is 2b+3, while the numerator on the right-hand side is 3b. These are not equal, so option B is not correct.
  • Option C: This option is equivalent to the original equation. The numerator on the left-hand side is b-4, while the numerator on the right-hand side is 3. These are equal, so option C is correct.

Q&A

Q: What is the purpose of rewriting an equation as a proportion? A: The purpose of rewriting an equation as a proportion is to simplify the equation and make it easier to solve.

Q: How do I rewrite an equation as a proportion? A: To rewrite an equation as a proportion, you need to isolate the fractions on one side of the equation and then multiply both sides by the reciprocal of the denominator.

Q: What is the difference between a proportion and an equation? A: A proportion is a statement that two ratios are equal, while an equation is a statement that two expressions are equal.

Q: How do I determine if a proportion is equivalent to an equation? A: To determine if a proportion is equivalent to an equation, you need to check if the numerators and denominators of the two ratios are equal.

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 isolating the fractions on one side of the equation
  • Not multiplying both sides by the reciprocal of the denominator
  • Not checking if the numerators and denominators of the two ratios are equal

Conclusion

In conclusion, rewriting an equation as a proportion is a useful technique for simplifying and solving equations. By following the steps outlined in this article, you can rewrite an equation as a proportion and determine if it is equivalent to the original equation.