Problem 13 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 5(x-7)+4(x+2) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(9x - 27\).

Step 1: Distribute Terms

First, distribute the terms outside of the parentheses to the inside of the parentheses. So \(5(x - 7)\) is equivalent to \(5x - 35\) and \(4(x + 2)\) is equivalent to \(4x + 8\). The expression can thus be rewritten as \(5x - 35 + 4x + 8\)

Step 2: Combine Like Terms

Now, the like terms in the expression, namely the 'x' terms and the constants, need to be combined. Add \(5x\) and \(4x\) to get \(9x\). Then, add \(-35\) and \(8\) to get \(-27\). The simplified expression is therefore \(9x - 27\)

Key Concepts

Distributive Property

Understanding the distributive property is essential in simplifying algebraic expressions. It allows you to remove parentheses by distributing a multiplication over addition or subtraction within the parentheses. For example, in the expression 5(x - 7) + 4(x + 2), we apply the distributive property by multiplying each term inside the parentheses by the number outside. So, 5(x - 7) becomes 5x - 35, and 4(x + 2) becomes 4x + 8. The property is based on the rule: a(b + c) = ab + ac.

Using this property accurately is crucial to ensure that every term is correctly accounted for before moving onto subsequent steps such as combining like terms. When it comes to mathematical operations, careful attention to each step helps prevent errors and solidifies understanding.

• Remember to multiply each term inside the parentheses by the term outside.
• Apply this property regardless of whether the operation inside the parentheses is addition or subtraction.
• After distributing, always double-check your work to ensure no terms have been omitted.

Combining Like Terms

Once the distributive property is applied and the algebraic expression is expanded, the next step is combining like terms. Like terms are terms that have the same variables raised to the same power. In the expression 5x - 35 + 4x + 8, the like terms are 5x and 4x, as well as -35 and 8, which are constants.

To simplify the expression further, you add or subtract the coefficients of the like terms. Adding 5x and 4x results in 9x because you're combining the 'x' terms. Similarly, combining -35 with 8 gives -27, which simplifies the constant part of the expression.

• Group like terms together before combining them.
• Add or subtract only the coefficients and keep the common variable part intact.
• After combining, ensure no terms have been overlooked.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. For instance, 9x - 27, the simplified result of our initial problem, is an algebraic expression. Understanding how to work with these expressions is key to mastering algebra. They can be simplified, evaluated, factored, or expanded based on the principles of algebra.

When simplifying expressions, pay attention to every component: variables, coefficients, constants, and the operations that connect them. Every expression tells a mathematical story, and simplifying it makes the story easier to understand.

• Algebraic expressions can take various forms, including polynomials, rational expressions, or even more complex relationships.
• Knowledge of operation rules, such as the distributive property and combining like terms, is critical in manipulating these expressions.
• Regular practice in simplifying algebraic expressions helps in building proficiency and confidence in algebra.