What Is The Sum In Simplest Form? 5 X + 3 10 + 3 X − 1 2 \frac{5x+3}{10}+\frac{3x-1}{2} 10 5 X + 3 ​ + 2 3 X − 1 ​

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Introduction

In mathematics, simplifying expressions is a crucial step in solving equations and inequalities. When dealing with fractions, combining them can be a bit challenging, but with the right approach, it can be done efficiently. In this article, we will explore how to simplify the sum of two fractions, $\frac{5x+3}{10}+\frac{3x-1}{2}$, and express it in its simplest form.

Understanding the Problem

To simplify the given expression, we need to first find a common denominator for the two fractions. The common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are 10 and 2. The LCM of 10 and 2 is 10.

Finding the Common Denominator

The common denominator is 10. To make the denominators of both fractions equal to 10, we need to multiply the numerator and denominator of the second fraction by 5.

$\frac{3x-1}{2} \cdot \frac{5}{5} = \frac{15x-5}{10}$

Combining the Fractions

Now that both fractions have the same denominator, we can combine them by adding the numerators.

$\frac{5x+3}{10} + \frac{15x-5}{10} = \frac{(5x+3)+(15x-5)}{10}$

Simplifying the Expression

To simplify the expression, we need to combine like terms in the numerator.

$\frac{(5x+3)+(15x-5)}{10} = \frac{20x-2}{10}$

Simplifying Further

We can simplify the expression further by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 20 and 10 is 10.

$\frac{20x-2}{10} = \frac{2(10x-1)}{10}$

Final Simplification

We can simplify the expression further by canceling out the common factor of 2 in the numerator and denominator.

$\frac{2(10x-1)}{10} = \frac{10x-1}{5}$

Conclusion

In conclusion, the sum of the two fractions, $\frac{5x+3}{10}+\frac{3x-1}{2}$, in its simplest form is $\frac{10x-1}{5}$. This expression is the result of combining the two fractions and simplifying the resulting expression.

Tips and Tricks

• When combining fractions, make sure to find a common denominator.
• Use the least common multiple (LCM) to find the common denominator.
• Multiply the numerator and denominator of each fraction by the necessary factor to make the denominators equal.
• Combine like terms in the numerator.
• Simplify the expression by dividing both the numerator and denominator by their greatest common divisor (GCD).
• Cancel out common factors in the numerator and denominator.

Common Mistakes to Avoid

• Failing to find a common denominator.
• Not multiplying the numerator and denominator of each fraction by the necessary factor to make the denominators equal.
• Not combining like terms in the numerator.
• Not simplifying the expression by dividing both the numerator and denominator by their greatest common divisor (GCD).
• Not canceling out common factors in the numerator and denominator.

Real-World Applications

Simplifying expressions is a crucial step in solving equations and inequalities in various fields, including physics, engineering, and economics. In physics, for example, simplifying expressions can help us understand the behavior of complex systems. In engineering, simplifying expressions can help us design and optimize systems. In economics, simplifying expressions can help us understand the behavior of markets and make informed decisions.

Final Thoughts

In conclusion, simplifying expressions is an essential skill in mathematics and has numerous real-world applications. By following the steps outlined in this article, you can simplify complex expressions and express them in their simplest form. Remember to find a common denominator, multiply the numerator and denominator of each fraction by the necessary factor, combine like terms, simplify the expression, and cancel out common factors. With practice and patience, you can become proficient in simplifying expressions and tackle complex problems with confidence.

Introduction

Simplifying expressions is a crucial step in solving equations and inequalities in mathematics. In our previous article, we explored how to simplify the sum of two fractions, $\frac{5x+3}{10}+\frac{3x-1}{2}$, and express it in its simplest form. In this article, we will answer some frequently asked questions about simplifying expressions.

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to find a common denominator for all the fractions in the expression. This is because fractions with different denominators cannot be added or subtracted directly.

Q: How do I find a common denominator?

A: To find a common denominator, you need to find the least common multiple (LCM) of all the denominators in the expression. The LCM is the smallest number that is a multiple of all the denominators.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you need to find the least common multiple (LCM) of the variable and the other denominators in the expression. This will give you the common denominator.

Q: How do I simplify an expression with multiple fractions?

A: To simplify an expression with multiple fractions, you need to follow these steps:

1. Find a common denominator for all the fractions.
2. Multiply the numerator and denominator of each fraction by the necessary factor to make the denominators equal.
3. Combine like terms in the numerator.
4. Simplify the expression by dividing both the numerator and denominator by their greatest common divisor (GCD).
5. Cancel out common factors in the numerator and denominator.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of an expression without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the Euclidean algorithm or simply list the factors of each number and find the largest common factor.

Q: What if I have a fraction with a negative exponent?

A: If you have a fraction with a negative exponent, you need to rewrite the fraction with a positive exponent by taking the reciprocal of the fraction.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to follow these steps:

1. Rewrite the fraction with a positive exponent by taking the reciprocal of the fraction.
2. Simplify the expression by combining like terms in the numerator.
3. Simplify the expression by dividing both the numerator and denominator by their greatest common divisor (GCD).
4. Cancel out common factors in the numerator and denominator.

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to cancel out any common factors in the numerator and denominator.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it helps us to:

• Understand complex expressions
• Solve equations and inequalities
• Make informed decisions in various fields, such as physics, engineering, and economics
• Communicate mathematical ideas clearly and effectively

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by:

• Working on math problems and exercises
• Using online resources and tools, such as calculators and software
• Joining a study group or working with a tutor
• Reading and studying math textbooks and resources

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can simplify complex expressions and express them in their simplest form. Remember to find a common denominator, multiply the numerator and denominator of each fraction by the necessary factor, combine like terms, simplify the expression, and cancel out common factors. With practice and patience, you can become proficient in simplifying expressions and tackle complex problems with confidence.