Fill In The Blank To Make Equivalent Rational Expressions.$\frac{u+6}{5-7u}=\frac{\square}{56u-40}$

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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.

Understanding Equivalent Rational Expressions

Equivalent rational expressions are expressions that have the same value for all values of the variable. For example, the expressions x+1x+2\frac{x+1}{x+2} and x+1x+2⋅x−1x−1\frac{x+1}{x+2} \cdot \frac{x-1}{x-1} are equivalent because they have the same value for all values of xx. To make two rational expressions equivalent, we need to multiply both the numerator and denominator of one expression by the same non-zero value.

Step 1: Identify the Non-Zero Value

To make the rational expressions u+65−7u\frac{u+6}{5-7u} and □56u−40\frac{\square}{56u-40} equivalent, we need to identify the non-zero value that we will multiply both the numerator and denominator of one expression by. In this case, we can see that the denominator of the second expression is 56u−4056u-40, which can be factored as 4(14u−10)4(14u-10). Therefore, the non-zero value that we will multiply both the numerator and denominator of one expression by is 44.

Step 2: Multiply the Numerator and Denominator

Now that we have identified the non-zero value, we can multiply both the numerator and denominator of one expression by this value. In this case, we will multiply the numerator and denominator of the first expression by 44. This gives us:

u+65−7u⋅44=4(u+6)4(5−7u)\frac{u+6}{5-7u} \cdot \frac{4}{4} = \frac{4(u+6)}{4(5-7u)}

Step 3: Simplify the Expression

Now that we have multiplied the numerator and denominator of one expression by the non-zero value, we can simplify the expression. In this case, we can simplify the expression by canceling out the common factor of 44 in the numerator and denominator. This gives us:

4(u+6)4(5−7u)=u+65−7u\frac{4(u+6)}{4(5-7u)} = \frac{u+6}{5-7u}

Step 4: Fill in the Blank

Now that we have simplified the expression, we can fill in the blank to make the two rational expressions equivalent. In this case, we can see that the numerator of the second expression is â–¡\square, which is equal to 4(u+6)4(u+6). Therefore, we can fill in the blank as follows:

u+65−7u=4(u+6)56u−40\frac{u+6}{5-7u} = \frac{4(u+6)}{56u-40}

Conclusion

In this article, we learned how to fill in the blank to make equivalent rational expressions. We identified the non-zero value that we would multiply both the numerator and denominator of one expression by, multiplied the numerator and denominator of one expression by this value, simplified the expression, and filled in the blank to make the two rational expressions equivalent. We hope that this article has been helpful in understanding how to make equivalent rational expressions.

Examples

Here are some examples of how to fill in the blank to make equivalent rational expressions:

  • x−2x+3=â–¡x2+6x+9\frac{x-2}{x+3} = \frac{\square}{x^2+6x+9}
  • y+42y−5=â–¡4y2−15y+20\frac{y+4}{2y-5} = \frac{\square}{4y^2-15y+20}
  • z−13z+2=â–¡9z2+6z+4\frac{z-1}{3z+2} = \frac{\square}{9z^2+6z+4}

Tips and Tricks

Here are some tips and tricks for filling in the blank to make equivalent rational expressions:

  • Identify the non-zero value that you will multiply both the numerator and denominator of one expression by.
  • Multiply the numerator and denominator of one expression by this value.
  • Simplify the expression by canceling out common factors.
  • Fill in the blank to make the two rational expressions equivalent.

Common Mistakes

Here are some common mistakes to avoid when filling in the blank to make equivalent rational expressions:

  • Not identifying the non-zero value that you will multiply both the numerator and denominator of one expression by.
  • Not multiplying the numerator and denominator of one expression by this value.
  • Not simplifying the expression by canceling out common factors.
  • Not filling in the blank to make the two rational expressions equivalent.

Real-World Applications

Here are some real-world applications of equivalent rational expressions:

  • In physics, equivalent rational expressions are used to describe the motion of objects.
  • In engineering, equivalent rational expressions are used to design and optimize systems.
  • In economics, equivalent rational expressions are used to model and analyze economic systems.

Conclusion

Q: What is an equivalent rational expression?

A: An equivalent rational expression is an expression that has the same value for all values of the variable. In other words, two rational expressions are equivalent if they have the same numerator and denominator, or if they can be made to have the same numerator and denominator by multiplying both the numerator and denominator by the same non-zero value.

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

A: To determine if two rational expressions are equivalent, you can try to simplify one of the expressions by canceling out common factors. If the simplified expression is equal to the other expression, then the two expressions are equivalent.

Q: What is the process for making two rational expressions equivalent?

A: The process for making two rational expressions equivalent involves the following steps:

  1. Identify the non-zero value that you will multiply both the numerator and denominator of one expression by.
  2. Multiply the numerator and denominator of one expression by this value.
  3. Simplify the expression by canceling out common factors.
  4. Fill in the blank to make the two rational expressions equivalent.

Q: What are some common mistakes to avoid when making two rational expressions equivalent?

A: Some common mistakes to avoid when making two rational expressions equivalent include:

  • Not identifying the non-zero value that you will multiply both the numerator and denominator of one expression by.
  • Not multiplying the numerator and denominator of one expression by this value.
  • Not simplifying the expression by canceling out common factors.
  • Not filling in the blank to make the two rational expressions equivalent.

Q: What are some real-world applications of equivalent rational expressions?

A: Equivalent rational expressions have many real-world applications, including:

  • In physics, equivalent rational expressions are used to describe the motion of objects.
  • In engineering, equivalent rational expressions are used to design and optimize systems.
  • In economics, equivalent rational expressions are used to model and analyze economic systems.

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

A: To know if you have made two rational expressions equivalent, you can try to simplify one of the expressions by canceling out common factors. If the simplified expression is equal to the other expression, then the two expressions are equivalent.

Q: What are some examples of equivalent rational expressions?

A: Some examples of equivalent rational expressions include:

  • x−2x+3=x−2x+3â‹…x−1x−1\frac{x-2}{x+3} = \frac{x-2}{x+3} \cdot \frac{x-1}{x-1}
  • y+42y−5=y+42y−5â‹…22\frac{y+4}{2y-5} = \frac{y+4}{2y-5} \cdot \frac{2}{2}
  • z−13z+2=z−13z+2â‹…33\frac{z-1}{3z+2} = \frac{z-1}{3z+2} \cdot \frac{3}{3}

Q: How do I fill in the blank to make two rational expressions equivalent?

A: To fill in the blank to make two rational expressions equivalent, you can follow these steps:

  1. Identify the non-zero value that you will multiply both the numerator and denominator of one expression by.
  2. Multiply the numerator and denominator of one expression by this value.
  3. Simplify the expression by canceling out common factors.
  4. Fill in the blank to make the two rational expressions equivalent.

Q: What are some tips and tricks for making two rational expressions equivalent?

A: Some tips and tricks for making two rational expressions equivalent include:

  • Identify the non-zero value that you will multiply both the numerator and denominator of one expression by.
  • Multiply the numerator and denominator of one expression by this value.
  • Simplify the expression by canceling out common factors.
  • Fill in the blank to make the two rational expressions equivalent.

Conclusion

In conclusion, equivalent rational expressions are an important concept in algebra. By understanding how to make two rational expressions equivalent, you can solve a wide range of problems in algebra. We hope that this article has been helpful in understanding how to make equivalent rational expressions.