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: $-11c - (4 - 2c$\]61. Subtract: $6x - 1$ From

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 a series of algebraic expression simplification problems.

Understanding Algebraic Expressions

An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. Algebraic expressions can be simple or complex, and they can be used to represent a wide range of mathematical concepts, from basic arithmetic operations to advanced calculus.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and eliminating parentheses. Like terms are terms that have the same variable raised to the same power. To simplify an algebraic expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying Algebraic Expressions with Parentheses

When simplifying algebraic expressions with parentheses, we need to follow the order of operations and evaluate the expressions inside the parentheses first.

Example 1: Simplifying $2(6x - 1) - (x - 7)$

To simplify this expression, we need to evaluate the expressions inside the parentheses first.

2(6x - 1) - (x - 7)
= 2(6x) - 2(1) - x + 7
= 12x - 2 - x + 7
= 11x + 5

Example 2: Simplifying $-2(3x - 4) + 7x - 6$

To simplify this expression, we need to evaluate the expressions inside the parentheses first.

-2(3x - 4) + 7x - 6
= -2(3x) + 2(4) + 7x - 6
= -6x + 8 + 7x - 6
= x + 2

Example 3: Simplifying $8y - 2 - 3(y + 4)$

To simplify this expression, we need to evaluate the expressions inside the parentheses first.

8y - 2 - 3(y + 4)
= 8y - 2 - 3y - 12
= 5y - 14

Example 4: Simplifying $5k - (3k - 10)$

To simplify this expression, we need to evaluate the expressions inside the parentheses first.

5k - (3k - 10)
= 5k - 3k + 10
= 2k + 10

Example 5: Simplifying $-11c - (4 - 2c)$

To simplify this expression, we need to evaluate the expressions inside the parentheses first.

-11c - (4 - 2c)
= -11c - 4 + 2c
= -9c - 4

Conclusion

Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the order of operations and evaluating expressions inside parentheses first, we can simplify complex algebraic expressions and make them easier to work with. In this article, we have provided step-by-step solutions to a series of algebraic expression simplification problems, and we have highlighted the importance of understanding algebraic expressions and simplifying them using the rules and techniques of algebra.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

1. Simplify: $3(2x + 1) - (x - 2)$
2. Simplify: $-4(2x - 3) + 5x - 1$
3. Simplify: $6y - 2 - 2(y + 3)$
4. Simplify: $4k - (2k - 5)$
5. Simplify: $-3c - (2 - c)$

Answer Key

1. $5x + 5$
2. $-3x + 11$
3. $4y - 8$
4. $2k + 5$
5. $-3c - 2$

Q: What is the order of operations in simplifying algebraic expressions?

A: The order of operations in simplifying algebraic expressions is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression with parentheses?

A: To simplify an algebraic expression with parentheses, you need to evaluate the expressions inside the parentheses first. This involves following the order of operations and simplifying the expressions inside the parentheses.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms in an algebraic expression, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $3x + 2x$, you can combine the like terms by adding the coefficients: $3x + 2x = 5x$.

Q: What is the distributive property in algebra?

A: The distributive property in algebra is a rule that allows you to multiply a single term by multiple terms inside parentheses. For example, if you have the expression $2(3x + 4)$, you can use the distributive property to multiply the single term $2$ by each term inside the parentheses: $2(3x + 4) = 6x + 8$.

Q: How do I simplify an algebraic expression with multiple sets of parentheses?

A: To simplify an algebraic expression with multiple sets of parentheses, you need to follow the order of operations and evaluate the expressions inside the parentheses from left to right. For example, if you have the expression $2(3x + 4) - (x - 2)$, you can simplify it by following the order of operations: $2(3x + 4) - (x - 2) = 6x + 8 - x + 2 = 5x + 10$.

Q: What is the difference between a variable and a constant in algebra?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

Q: How do I simplify an algebraic expression with variables and constants?

A: To simplify an algebraic expression with variables and constants, you need to follow the order of operations and combine like terms. For example, if you have the expression $3x + 2 + 4x$, you can simplify it by combining the like terms: $3x + 2 + 4x = 7x + 2$.

Q: What is the final answer to the expression $2(6x - 1) - (x - 7)$?

A: The final answer to the expression $2(6x - 1) - (x - 7)$ is $11x + 5$.

Q: What is the final answer to the expression $-2(3x - 4) + 7x - 6$?

A: The final answer to the expression $-2(3x - 4) + 7x - 6$ is $x + 2$.

Q: What is the final answer to the expression $8y - 2 - 3(y + 4)$?

A: The final answer to the expression $8y - 2 - 3(y + 4)$ is $5y - 14$.

Q: What is the final answer to the expression $5k - (3k - 10)$?

A: The final answer to the expression $5k - (3k - 10)$ is $2k + 10$.

Q: What is the final answer to the expression $-11c - (4 - 2c)$?

A: The final answer to the expression $-11c - (4 - 2c)$ is $-9c - 4$.

By following these frequently asked questions and answers, you can gain a better understanding of simplifying algebraic expressions and become proficient in solving algebraic problems.