Subtract The Following Expressions And Simplify Your Answer As Much As Possible:$\[ \frac{5x-3}{3x} - \frac{4x+1}{5x} \\]

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Introduction

When dealing with algebraic expressions, it's often necessary to perform operations such as addition, subtraction, multiplication, and division. In this article, we will focus on subtracting two algebraic expressions and simplifying the result. We will use the given expressions 5x−33x−4x+15x\frac{5x-3}{3x} - \frac{4x+1}{5x} as an example and demonstrate the step-by-step process of subtracting and simplifying the expressions.

Understanding the Problem

To begin, let's examine the given expressions and understand what is being asked. We are given two algebraic expressions, 5x−33x\frac{5x-3}{3x} and 4x+15x\frac{4x+1}{5x}, and we are asked to subtract the second expression from the first expression. This means we need to find a common denominator and then perform the subtraction.

Finding a Common Denominator

To subtract the two expressions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two expressions. In this case, the denominators are 3x3x and 5x5x. The LCM of 3x3x and 5x5x is 15x15x.

Subtracting the Expressions

Now that we have found the common denominator, we can subtract the two expressions. We will rewrite each expression with the common denominator and then perform the subtraction.

5x−33x−4x+15x=5x−33x⋅55−4x+15x⋅33\frac{5x-3}{3x} - \frac{4x+1}{5x} = \frac{5x-3}{3x} \cdot \frac{5}{5} - \frac{4x+1}{5x} \cdot \frac{3}{3}

=25x−1515x−12x+315x= \frac{25x-15}{15x} - \frac{12x+3}{15x}

Combining the Fractions

Now that we have rewritten the expressions with the common denominator, we can combine the fractions by subtracting the numerators.

25x−1515x−12x+315x=(25x−15)−(12x+3)15x\frac{25x-15}{15x} - \frac{12x+3}{15x} = \frac{(25x-15) - (12x+3)}{15x}

=25x−15−12x−315x= \frac{25x-15-12x-3}{15x}

Simplifying the Expression

Now that we have combined the fractions, we can simplify the expression by combining like terms in the numerator.

25x−15−12x−315x=13x−1815x\frac{25x-15-12x-3}{15x} = \frac{13x-18}{15x}

Final Answer

The final answer is 13x−1815x\frac{13x-18}{15x}. This is the simplified result of subtracting the two given expressions.

Conclusion

In this article, we demonstrated the step-by-step process of subtracting two algebraic expressions and simplifying the result. We used the given expressions 5x−33x−4x+15x\frac{5x-3}{3x} - \frac{4x+1}{5x} as an example and found the common denominator, subtracted the expressions, combined the fractions, and simplified the result. The final answer is 13x−1815x\frac{13x-18}{15x}.

Additional Tips and Tricks

  • When subtracting algebraic expressions, it's essential to find a common denominator to ensure that the expressions are being subtracted correctly.
  • When combining fractions, make sure to subtract the numerators and keep the common denominator.
  • When simplifying expressions, look for opportunities to combine like terms in the numerator.

Frequently Asked Questions

  • Q: What is the common denominator of the two expressions? A: The common denominator is the least common multiple (LCM) of the denominators of the two expressions. In this case, the LCM of 3x3x and 5x5x is 15x15x.
  • Q: How do I subtract the expressions? A: To subtract the expressions, rewrite each expression with the common denominator and then perform the subtraction.
  • Q: How do I simplify the expression? A: To simplify the expression, combine like terms in the numerator.

References

Introduction

Subtracting algebraic expressions can be a challenging task, especially when dealing with complex expressions. In this article, we will address some of the most frequently asked questions related to subtracting algebraic expressions. We will provide detailed explanations and examples to help you understand the concepts better.

Q&A

Q: What is the first step in subtracting algebraic expressions?

A: The first step in subtracting algebraic expressions is to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two expressions.

Q: How do I find the common denominator?

A: To find the common denominator, you need to identify the denominators of the two expressions and then find their LCM. For example, if the denominators are 3x3x and 5x5x, the LCM is 15x15x.

Q: What if the denominators are not the same?

A: If the denominators are not the same, you need to find the LCM of the two denominators. For example, if the denominators are 2x2x and 3x3x, the LCM is 6x6x.

Q: How do I subtract the expressions?

A: To subtract the expressions, rewrite each expression with the common denominator and then perform the subtraction. For example, if the expressions are 5x−33x\frac{5x-3}{3x} and 4x+15x\frac{4x+1}{5x}, you would rewrite them as 5x−33x⋅55\frac{5x-3}{3x} \cdot \frac{5}{5} and 4x+15x⋅33\frac{4x+1}{5x} \cdot \frac{3}{3}, respectively.

Q: What if the expressions have different variables?

A: If the expressions have different variables, you need to find a common variable and then rewrite the expressions in terms of that variable. For example, if the expressions are 5x−33x\frac{5x-3}{3x} and 4y+15y\frac{4y+1}{5y}, you would rewrite them as 5x−33x⋅yy\frac{5x-3}{3x} \cdot \frac{y}{y} and 4y+15y⋅xx\frac{4y+1}{5y} \cdot \frac{x}{x}, respectively.

Q: How do I simplify the expression?

A: To simplify the expression, combine like terms in the numerator. For example, if the expression is 13x−1815x\frac{13x-18}{15x}, you would combine the like terms to get 13x−1815x=13x15x−1815x\frac{13x-18}{15x} = \frac{13x}{15x} - \frac{18}{15x}.

Q: What if the expression has a negative sign?

A: If the expression has a negative sign, you need to distribute the negative sign to the terms in the numerator. For example, if the expression is −13x−1815x-\frac{13x-18}{15x}, you would distribute the negative sign to get −13x15x+1815x-\frac{13x}{15x} + \frac{18}{15x}.

Q: Can I subtract algebraic expressions with different coefficients?

A: Yes, you can subtract algebraic expressions with different coefficients. For example, if the expressions are 2x2+3x2x^2 + 3x and x2+2xx^2 + 2x, you would subtract them as follows: (2x2+3x)−(x2+2x)=2x2−x2+3x−2x=x2+x(2x^2 + 3x) - (x^2 + 2x) = 2x^2 - x^2 + 3x - 2x = x^2 + x.

Q: Can I subtract algebraic expressions with different exponents?

A: Yes, you can subtract algebraic expressions with different exponents. For example, if the expressions are x2+3xx^2 + 3x and 2x3+x2x^3 + x, you would subtract them as follows: (x2+3x)−(2x3+x)=x2−2x3+3x−x=−2x3+x2+2x(x^2 + 3x) - (2x^3 + x) = x^2 - 2x^3 + 3x - x = -2x^3 + x^2 + 2x.

Conclusion

Subtracting algebraic expressions can be a challenging task, but with the right techniques and strategies, you can simplify complex expressions and solve problems with ease. Remember to find a common denominator, rewrite the expressions with the common denominator, and then perform the subtraction. Finally, simplify the expression by combining like terms in the numerator.

Additional Tips and Tricks

  • When subtracting algebraic expressions, make sure to find a common denominator to ensure that the expressions are being subtracted correctly.
  • When combining fractions, make sure to subtract the numerators and keep the common denominator.
  • When simplifying expressions, look for opportunities to combine like terms in the numerator.
  • When dealing with different coefficients or exponents, make sure to distribute the negative sign and combine like terms accordingly.

Frequently Asked Questions

  • Q: What is the first step in subtracting algebraic expressions? A: The first step in subtracting algebraic expressions is to find a common denominator.
  • Q: How do I find the common denominator? A: To find the common denominator, you need to identify the denominators of the two expressions and then find their LCM.
  • Q: What if the denominators are not the same? A: If the denominators are not the same, you need to find the LCM of the two denominators.

References