(-8/7t + 4/12)-(-3/14t + 7/4)
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying a complex algebraic expression involving fractions and variables. We will use the expression (-8/7t + 4/12) - (-3/14t + 7/4) as a case study and provide a step-by-step guide on how to simplify it.
Understanding the Expression
The given expression is a combination of two fractions with variables and constants. To simplify it, we need to understand the rules of algebraic operations, including the order of operations (PEMDAS) and the rules for adding and subtracting fractions.
PEMDAS: A Review
Before we dive into simplifying the expression, let's review the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Simplifying the Expression
Now that we have a good understanding of the expression and the rules of algebraic operations, let's simplify it step by step.
Step 1: Distribute the Negative Sign
The first step in simplifying the expression is to distribute the negative sign to the terms inside the second parentheses.
(-8/7t + 4/12) - (-3/14t + 7/4) = (-8/7t + 4/12) + (3/14t - 7/4)
Step 2: Find a Common Denominator
To add and subtract fractions, we need to find a common denominator. The least common multiple (LCM) of 12 and 14 is 84.
(-8/7t + 4/12) + (3/14t - 7/4) = (-8/7t + 34/84) + (21/84t - 7/4)
Step 3: Add and Subtract Fractions
Now that we have a common denominator, we can add and subtract the fractions.
(-8/7t + 34/84) + (21/84t - 7/4) = (-8/7t + 21/84t) + (34/84 - 7/4)
Step 4: Simplify the Fractions
To simplify the fractions, we need to find the least common multiple (LCM) of 7 and 84, which is 84. We can also simplify the second fraction by finding the LCM of 84 and 4, which is 84.
(-8/7t + 21/84t) + (34/84 - 7/4) = (-48/84t + 21/84t) + (34/84 - 7/4)
Step 5: Combine Like Terms
Now that we have simplified the fractions, we can combine like terms.
(-48/84t + 21/84t) + (34/84 - 7/4) = (-27/84t) + (34/84 - 7/4)
Step 6: Simplify the Second Fraction
To simplify the second fraction, we need to find the LCM of 84 and 4, which is 84.
(-27/84t) + (34/84 - 7/4) = (-27/84t) + (34/84 - 7/12)
Step 7: Find a Common Denominator
To add and subtract fractions, we need to find a common denominator. The LCM of 84 and 12 is 84.
(-27/84t) + (34/84 - 7/12) = (-27/84t) + (34/84 - 49/84)
Step 8: Add and Subtract Fractions
Now that we have a common denominator, we can add and subtract the fractions.
(-27/84t) + (34/84 - 49/84) = (-27/84t) + (-15/84)
Step 9: Simplify the Expression
Finally, we can simplify the expression by combining like terms.
(-27/84t) + (-15/84) = (-42/84t)
Conclusion
Simplifying algebraic expressions is a crucial skill for students and professionals alike. In this article, we used the expression (-8/7t + 4/12) - (-3/14t + 7/4) as a case study and provided a step-by-step guide on how to simplify it. By following the rules of algebraic operations and simplifying fractions, we were able to simplify the expression to (-42/84t).
Final Answer
The final answer is:
(-42/84t)
Simplifying Further
We can simplify the expression further by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 6.
(-42/84t) = (-7/14t)
Final Answer
The final answer is:
(-7/14t)
Real-World Applications
Simplifying algebraic expressions has numerous real-world applications, including:
- Physics: Simplifying algebraic expressions is crucial in physics, where equations often involve complex variables and constants.
- Engineering: Simplifying algebraic expressions is essential in engineering, where equations often involve complex variables and constants.
- Computer Science: Simplifying algebraic expressions is a fundamental concept in computer science, where algorithms often involve complex variables and constants.
Conclusion
In conclusion, simplifying algebraic expressions is a crucial skill for students and professionals alike. By following the rules of algebraic operations and simplifying fractions, we can simplify complex expressions and solve real-world problems.
Introduction
Simplifying algebraic expressions is a crucial skill for students and professionals alike. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.
Q: What is an algebraic expression?
A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
Q: Why is it important to simplify algebraic expressions?
A: Simplifying algebraic expressions is important because it helps to:
- Reduce complexity: Simplifying algebraic expressions reduces the complexity of the expression, making it easier to understand and work with.
- Improve accuracy: Simplifying algebraic expressions helps to eliminate errors and improve accuracy.
- Solve problems efficiently: Simplifying algebraic expressions helps to solve problems efficiently and effectively.
Q: What are the steps to simplify an algebraic expression?
A: The steps to simplify an algebraic expression are:
- Distribute the negative sign: Distribute the negative sign to the terms inside the parentheses.
- Find a common denominator: Find a common denominator for the fractions.
- Add and subtract fractions: Add and subtract the fractions.
- Simplify the fractions: Simplify the fractions by dividing both the numerator and the denominator by their greatest common divisor (GCD).
- Combine like terms: Combine like terms.
Q: How do I simplify a fraction?
A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Then, divide both the numerator and the denominator by the GCD.
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when working with mathematical expressions. The order of operations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I simplify an expression with multiple variables?
A: To simplify an expression with multiple variables, you need to follow the same steps as before, but you also need to consider the relationships between the variables.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not distributing the negative sign: Not distributing the negative sign to the terms inside the parentheses.
- Not finding a common denominator: Not finding a common denominator for the fractions.
- Not simplifying the fractions: Not simplifying the fractions by dividing both the numerator and the denominator by their greatest common divisor (GCD).
- Not combining like terms: Not combining like terms.
Q: How do I check my work when simplifying algebraic expressions?
A: To check your work when simplifying algebraic expressions, you can:
- Plug in values: Plug in values for the variables to see if the expression simplifies correctly.
- Use a calculator: Use a calculator to check the expression.
- Check the units: Check the units to make sure they are correct.
Conclusion
Simplifying algebraic expressions is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can simplify complex expressions and solve real-world problems. Remember to avoid common mistakes and check your work to ensure accuracy.