Problem 3 - Algebra 1: Concepts and Skills

Question

Explain how you would use the distributive property to simplify the expression. $$ 7(x-3) $$

Step-by-Step Solution

Answer

The simplified form of the expression \(7(x - 3)\) is \(7x - 21\).

Step 1: Apply the Distributive Property

To simplify the expression \(7(x-3)\) using the distributive property, apply '7' to the terms inside the parentheses. The distributive property states that for all real numbers 'a', 'b', and 'c': \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\). So, multiply '7' with 'x' and also '7' with '-3'.

Step 2: Perform the Multiplication

Performing the multiplication gives \(7x - 21\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial part of solving and mastering algebraic problems. When we simplify an expression, we aim to make it easier to understand or work with by transforming it into a simpler form. This often involves eliminating parentheses, combining like terms, and ensuring each term is in its simplest form.
For example, in the expression \(7(x-3)\), simplifying means using techniques like the distributive property to eliminate parentheses and reduce the expression to a single mathematical statement without changing its value.

• Eliminate parentheses using properties such as distributive or associative.
• Combine like terms, which are terms that have the same variables raised to the same power.
• Reduce the terms if possible, for instance by factoring or using identities.

When done correctly, simplifying can reveal more insights into what the expression represents, making it easier to solve equations or make calculations.

Algebraic Expressions

Algebraic expressions may look intimidating, but they are simply mathematical phrases that can include numbers, variables, and operations. Variables, often represented by letters like 'x' or 'y', are placeholders for numbers or quantities that can change.
For example, \(7(x-3)\) includes a number '7', a variable 'x', and an operation subtracting '3'. These expressions allow for more flexibility and generality in solving mathematical problems as they can solve for many possible numbers, not just one set value.

• Recognize variables as symbols that can take different values.
• Understand operations within expressions, which include addition, subtraction, multiplication, and division.
• Know that expressions do not include an equal sign, which distinguishes them from equations.

Understanding algebraic expressions is key to learning algebra, as they form the basis for equations, inequalities, and many other algebraic concepts.

Multiplication in Algebra

Multiplication in algebra isn't very different from basic arithmetic multiplication; however, it becomes slightly more complex when variables are involved. In instances like \(7(x-3)\), we use the multiplication not just on numbers but extend it to include multiplication of terms with variables. This is where distributive property, a key multiplication strategy, becomes essential to break down and solve such expressions.
The principles of multiplication in algebra include:

• Applying numbers to variables, understanding that \(a \cdot x = ax\).
• Ensuring the correct application of associative, commutative, and distributive properties to simplify expressions.
• Remembering that multiplying by a negative number inverses the sign, like in \(7 \times (-3) = -21\).

Good grasp of multiplication in algebra will significantly aid in tackling more complex algebraic manipulations and expression simplifications.