55. Simplify: $3(2x - 5) - 5(x - 4$\]56. Simplify: $2(6x - 1) - (x - 7$\]57. Simplify: $-2(3x - 4) + 7x - 6$58. Simplify: $8y - 2 - 3(y + 4$\]59. Simplify: $5k - (3k - 10$\]60. Simplify: $-11c - (4 -
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will explore the process of simplifying algebraic expressions, focusing on the rules and techniques used to combine like terms and eliminate parentheses. We will also provide step-by-step solutions to 10 algebraic expression simplification problems, covering various scenarios and techniques.
Understanding Algebraic Expressions
An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It can be a single term or a combination of terms, connected by addition, subtraction, multiplication, or division. Algebraic expressions can be simplified by combining like terms, eliminating parentheses, and rearranging the terms.
Simplifying Algebraic Expressions: Rules and Techniques
To simplify algebraic expressions, we need to follow certain rules and techniques:
- Distributive Property: The distributive property states that for any real numbers a, b, and c, a(b + c) = ab + ac.
- Combining Like Terms: Like terms are terms that have the same variable raised to the same power. We can combine like terms by adding or subtracting their coefficients.
- Eliminating Parentheses: To eliminate parentheses, we need to distribute the terms inside the parentheses to the terms outside the parentheses.
- Rearranging Terms: We can rearrange the terms in an algebraic expression to make it easier to simplify.
Simplifying Algebraic Expressions: Step-by-Step Solutions
Problem 55: Simplify:
To simplify this expression, we need to follow the distributive property and combine like terms.
$3(2x - 5) - 5(x - 4)$
= $6x - 15 - 5x + 20$
= $x + 5$
Problem 56: Simplify:
To simplify this expression, we need to follow the distributive property and combine like terms.
$2(6x - 1) - (x - 7)$
= $12x - 2 - x + 7$
= $11x + 5$
Problem 57: Simplify:
To simplify this expression, we need to follow the distributive property and combine like terms.
$-2(3x - 4) + 7x - 6$
= $-6x + 8 + 7x - 6$
= $x + 2$
Problem 58: Simplify:
To simplify this expression, we need to follow the distributive property and combine like terms.
$8y - 2 - 3(y + 4)$
= $8y - 2 - 3y - 12$
= $5y - 14$
Problem 59: Simplify:
To simplify this expression, we need to follow the distributive property and combine like terms.
$5k - (3k - 10)$
= $5k - 3k + 10$
= $2k + 10$
Problem 60: Simplify:
To simplify this expression, we need to follow the distributive property and combine like terms.
$-11c - (4 - 3c)$
= $-11c - 4 + 3c$
= $-8c - 4$
Conclusion
Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the rules and techniques outlined in this article, we can simplify complex algebraic expressions and make them easier to work with. Remember to always follow the distributive property, combine like terms, eliminate parentheses, and rearrange terms to make the expression easier to simplify.
Final Tips
- Practice, practice, practice: The more you practice simplifying algebraic expressions, the more comfortable you will become with the rules and techniques.
- Use online resources: There are many online resources available that can help you simplify algebraic expressions, including calculators and online tools.
- Ask for help: If you are struggling to simplify an algebraic expression, don't be afraid to ask for help. Your teacher, tutor, or classmate may be able to provide guidance and support.
Q: What is the distributive property, and how is it used in simplifying algebraic expressions?
A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, a(b + c) = ab + ac. This property is used to simplify algebraic expressions by distributing the terms inside the parentheses to the terms outside the parentheses.
Q: How do I combine like terms in an algebraic expression?
A: To combine like terms, you need to identify the terms that have the same variable raised to the same power. Then, you can add or subtract their coefficients to simplify the expression.
Q: What is the difference between a like term and a unlike term?
A: A like term is a term that has the same variable raised to the same power. A unlike term is a term that has a different variable or a different power of the variable.
Q: How do I eliminate parentheses in an algebraic expression?
A: To eliminate parentheses, you need to distribute the terms inside the parentheses to the terms outside the parentheses using the distributive property.
Q: What is the order of operations in simplifying algebraic expressions?
A: The order of operations in simplifying algebraic expressions is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any 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 algebraic expression with multiple variables?
A: To simplify an algebraic expression with multiple variables, you need to follow the same rules and techniques as before. However, you need to be careful to combine like terms and eliminate parentheses correctly.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Forgetting to distribute terms inside parentheses.
- Failing to combine like terms.
- Not eliminating parentheses correctly.
- Making errors in the order of operations.
Q: How can I practice simplifying algebraic expressions?
A: You can practice simplifying algebraic expressions by working through exercises and problems in your textbook or online resources. You can also try simplifying expressions on your own and then checking your work with a calculator or online tool.
Q: What are some real-world applications of simplifying algebraic expressions?
A: Simplifying algebraic expressions has many real-world applications, including:
- Solving systems of equations in physics and engineering.
- Modeling population growth and decay in biology.
- Analyzing financial data in economics.
- Solving optimization problems in computer science.
By following these tips and practicing regularly, you will become proficient in simplifying algebraic expressions and be able to tackle even the most complex problems with confidence.