Section 2.1: Simplifying Algebraic Expressions70. $0.2(k+8)-0.1k$71. $10-3(2x+3y$\]72. $14-11(5m+3n$\]73. $6(3x-6)-2(x+1)-17x$74. $7(2x+5)-4(x+2)-20x$75. $\frac{1}{2}(12x-4)-(x+5$\]76.

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. 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 examples to illustrate the concepts.

Simplifying Expressions with Parentheses

When simplifying expressions with parentheses, 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 multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Example 1: Simplifying $0.2(k+8)-0.1k$

To simplify the expression $0.2(k+8)-0.1k$, we need to follow the order of operations:

1. Evaluate the expression inside the parentheses: $k+8$
2. Multiply $0.2$ by the result: $0.2(k+8) = 0.2k + 1.6$
3. Subtract $0.1k$ from the result: $0.2k + 1.6 - 0.1k = 0.1k + 1.6$

The simplified expression is $0.1k + 1.6$.

Example 2: Simplifying $10-3(2x+3y)$

To simplify the expression $10-3(2x+3y)$, we need to follow the order of operations:

1. Evaluate the expression inside the parentheses: $2x+3y$
2. Multiply $3$ by the result: $3(2x+3y) = 6x + 9y$
3. Subtract the result from $10$: $10 - 6x - 9y$

The simplified expression is $10 - 6x - 9y$.

Example 3: Simplifying $14-11(5m+3n)$

To simplify the expression $14-11(5m+3n)$, we need to follow the order of operations:

1. Evaluate the expression inside the parentheses: $5m+3n$
2. Multiply $11$ by the result: $11(5m+3n) = 55m + 33n$
3. Subtract the result from $14$: $14 - 55m - 33n$

The simplified expression is $14 - 55m - 33n$.

Example 4: Simplifying $6(3x-6)-2(x+1)-17x$

To simplify the expression $6(3x-6)-2(x+1)-17x$, we need to follow the order of operations:

1. Evaluate the expression inside the parentheses: $3x-6$
2. Multiply $6$ by the result: $6(3x-6) = 18x - 36$
3. Multiply $2$ by the result: $2(x+1) = 2x + 2$
4. Subtract the result from $18x - 36$: $18x - 36 - 2x - 2 = 16x - 38$
5. Subtract $17x$ from the result: $16x - 38 - 17x = -x - 38$

The simplified expression is $-x - 38$.

Example 5: Simplifying $7(2x+5)-4(x+2)-20x$

To simplify the expression $7(2x+5)-4(x+2)-20x$, we need to follow the order of operations:

1. Evaluate the expression inside the parentheses: $2x+5$
2. Multiply $7$ by the result: $7(2x+5) = 14x + 35$
3. Multiply $4$ by the result: $4(x+2) = 4x + 8$
4. Subtract the result from $14x + 35$: $14x + 35 - 4x - 8 = 10x + 27$
5. Subtract $20x$ from the result: $10x + 27 - 20x = -10x + 27$

The simplified expression is $-10x + 27$.

Example 6: Simplifying $\frac{1}{2}(12x-4)-(x+5)$

To simplify the expression $\frac{1}{2}(12x-4)-(x+5)$, we need to follow the order of operations:

1. Evaluate the expression inside the parentheses: $12x-4$
2. Multiply $\frac{1}{2}$ by the result: $\frac{1}{2}(12x-4) = 6x - 2$
3. Subtract $x$ from the result: $6x - 2 - x = 5x - 2$
4. Subtract $5$ from the result: $5x - 2 - 5 = 5x - 7$

The simplified expression is $5x - 7$.

Conclusion

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it can be a challenging task for students. In this article, we will provide a Q&A guide to help students understand the concepts and techniques used to simplify algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an algebraic expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate 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?

A: To simplify an algebraic expression, follow these steps:

1. Evaluate any expressions inside parentheses.
2. Combine like terms by adding or subtracting coefficients.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is a like term?

A: A like term is a term that has the same variable(s) and exponent(s) as another term. For example, 2x and 4x are like terms because they both have the variable x and the same exponent (1).

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x, and 3y - 2y = y.

Q: What is the distributive property?

A: The distributive property is a rule that allows us to multiply a single term by multiple terms. For example, 2(x + 3) = 2x + 6.

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, follow these steps:

1. Evaluate any expressions inside the parentheses.
2. Multiply the coefficient of the parentheses by the result.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a coefficient and a constant?

A: A coefficient is a number that is multiplied by a variable, while a constant is a number that is not multiplied by a variable. For example, in the expression 2x, 2 is the coefficient and x is the variable. In the expression 3, 3 is the constant.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, follow these steps:

1. Simplify any fractions in the expression.
2. Multiply the numerator and denominator of each fraction by the same number to eliminate any fractions.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Conclusion

Simplifying algebraic expressions is a crucial skill for students to master. By following the order of operations and using the rules for combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we have provided a Q&A guide to help students understand the concepts and techniques used to simplify algebraic expressions. With practice and patience, students can become proficient in simplifying expressions and solving algebraic problems.