Fill In The Blank To Make Equivalent Rational Expressions.$\frac{7v}{7v-6} = \frac{\square}{42v-36}$

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Introduction

In algebra, equivalent rational expressions are expressions that have the same value for all values of the variable. To make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same non-zero value. In this article, we will learn how to fill in the blank to make equivalent rational expressions.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator. For example, the rational expression 2x4x\frac{2x}{4x} can be simplified to x2x\frac{x}{2x} by canceling out the common factor of 2x2x.

How to Make Equivalent Rational Expressions

To make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same non-zero value. This value is called the equivalent ratio. The equivalent ratio is a fraction that has the same numerator and denominator as the original expression, but with the numerator and denominator swapped.

Example 1: Making Equivalent Rational Expressions

Let's consider the rational expression 7v7v−6\frac{7v}{7v-6}. We want to make this expression equivalent to □42v−36\frac{\square}{42v-36}. To do this, we need to find the equivalent ratio.

$\frac{7v}{7v-6} = \frac{\square}{42v-36}$

To find the equivalent ratio, we can multiply both the numerator and denominator of the first expression by 66. This gives us:

$\frac{7v}{7v-6} = \frac{7v \cdot 6}{(7v-6) \cdot 6}$

Simplifying the expression, we get:

$\frac{7v}{7v-6} = \frac{42v}{42v-36}$

Now, we can see that the numerator and denominator of the second expression are the same as the numerator and denominator of the first expression, but with the numerator and denominator swapped. Therefore, the equivalent ratio is 42v42v−36\frac{42v}{42v-36}.

Example 2: Making Equivalent Rational Expressions

Let's consider the rational expression x+2x+5\frac{x+2}{x+5}. We want to make this expression equivalent to â–¡x2+9x+20\frac{\square}{x^2+9x+20}. To do this, we need to find the equivalent ratio.

$\frac{x+2}{x+5} = \frac{\square}{x^2+9x+20}$

To find the equivalent ratio, we can multiply both the numerator and denominator of the first expression by x−2x-2. This gives us:

$\frac{x+2}{x+5} = \frac{(x+2)(x-2)}{(x+5)(x-2)}$

Simplifying the expression, we get:

$\frac{x+2}{x+5} = \frac{x^2-4}{x^2+x-10}$

Now, we can see that the numerator and denominator of the second expression are the same as the numerator and denominator of the first expression, but with the numerator and denominator swapped. Therefore, the equivalent ratio is x2−4x2+x−10\frac{x^2-4}{x^2+x-10}.

Conclusion

In this article, we learned how to fill in the blank to make equivalent rational expressions. We saw that to make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same non-zero value, called the equivalent ratio. We also saw that the equivalent ratio is a fraction that has the same numerator and denominator as the original expression, but with the numerator and denominator swapped. By following these steps, we can make two rational expressions equivalent.

Tips and Tricks

  • When making equivalent rational expressions, always multiply both the numerator and denominator of one expression by the same non-zero value.
  • The equivalent ratio is a fraction that has the same numerator and denominator as the original expression, but with the numerator and denominator swapped.
  • To find the equivalent ratio, you can multiply both the numerator and denominator of one expression by a common factor.

Practice Problems

  1. Make the rational expression 3x3x−2\frac{3x}{3x-2} equivalent to □9x2−6x\frac{\square}{9x^2-6x}.
  2. Make the rational expression x−1x+3\frac{x-1}{x+3} equivalent to □x2+2x−9\frac{\square}{x^2+2x-9}.
  3. Make the rational expression 2y2y+1\frac{2y}{2y+1} equivalent to â–¡4y2+2y\frac{\square}{4y^2+2y}.

Answer Key

  1. 9x2−6x9x2−6x\frac{9x^2-6x}{9x^2-6x}
  2. x2−4x2+2x−9\frac{x^2-4}{x^2+2x-9}
  3. 4y2+2y4y2+2y\frac{4y^2+2y}{4y^2+2y}
    Fill in the Blank to Make Equivalent Rational Expressions: Q&A ================================================================

Introduction

In our previous article, we learned how to fill in the blank to make equivalent rational expressions. We saw that to make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same non-zero value, called the equivalent ratio. In this article, we will answer some frequently asked questions about making equivalent rational expressions.

Q: What is the equivalent ratio?

A: The equivalent ratio is a fraction that has the same numerator and denominator as the original expression, but with the numerator and denominator swapped.

Q: How do I find the equivalent ratio?

A: To find the equivalent ratio, you can multiply both the numerator and denominator of one expression by a common factor.

Q: What if the numerator and denominator of the two expressions are not the same?

A: If the numerator and denominator of the two expressions are not the same, you can multiply both the numerator and denominator of one expression by a common factor to make them the same.

Q: Can I make a rational expression equivalent to itself?

A: Yes, you can make a rational expression equivalent to itself by multiplying both the numerator and denominator by 1.

Q: What if I make a mistake and multiply the numerator and denominator by the wrong value?

A: If you make a mistake and multiply the numerator and denominator by the wrong value, you can try again and multiply them by the correct value.

Q: Can I make a rational expression equivalent to a different type of expression, such as a polynomial or a fraction?

A: No, you can only make a rational expression equivalent to another rational expression.

Q: How do I know if two rational expressions are equivalent?

A: To know if two rational expressions are equivalent, you can multiply both the numerator and denominator of one expression by the same non-zero value and see if the resulting expression is the same as the other expression.

Q: Can I use a calculator to make equivalent rational expressions?

A: Yes, you can use a calculator to make equivalent rational expressions, but you should also try to do it by hand to understand the process.

Q: What if I get stuck and can't make the rational expressions equivalent?

A: If you get stuck and can't make the rational expressions equivalent, you can try asking a teacher or tutor for help.

Tips and Tricks

  • Always multiply both the numerator and denominator of one expression by the same non-zero value.
  • The equivalent ratio is a fraction that has the same numerator and denominator as the original expression, but with the numerator and denominator swapped.
  • To find the equivalent ratio, you can multiply both the numerator and denominator of one expression by a common factor.
  • You can use a calculator to make equivalent rational expressions, but you should also try to do it by hand to understand the process.

Practice Problems

  1. Make the rational expression 2x2x−1\frac{2x}{2x-1} equivalent to □4x2−2x\frac{\square}{4x^2-2x}.
  2. Make the rational expression x+1x−2\frac{x+1}{x-2} equivalent to □x2−3x−4\frac{\square}{x^2-3x-4}.
  3. Make the rational expression 3y3y+2\frac{3y}{3y+2} equivalent to â–¡9y2+6y\frac{\square}{9y^2+6y}.

Answer Key

  1. 4x2−2x4x2−2x\frac{4x^2-2x}{4x^2-2x}
  2. x2−3x−4x2−3x−4\frac{x^2-3x-4}{x^2-3x-4}
  3. 9y2+6y9y2+6y\frac{9y^2+6y}{9y^2+6y}