Problem 37 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 7(4 x-3) $$

Step-by-Step Solution

Answer

The expression simplifies to \( 28x - 21 \).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers or expressions \( a, b, \) and \( c, \) the equation \( a(b + c) = ab + ac \) holds. In this exercise, you have \( 7(4x - 3) \). Use the distributive property to multiply 7 by each term inside the parenthesis: \( 7 \times 4x \) and \( 7 \times -3 \).

Step 2: Perform the Multiplications

Carry out the multiplications from Step 1: \( 7 \times 4x = 28x \) and \( 7 \times -3 = -21 \). This gives us \( 28x - 21 \).

Step 3: Simplify the Expression

Examine the expression \( 28x - 21 \). There are no like terms to combine or further simplifications, so this is the final form.

Key Concepts

Multiplication

The multiplication step is essential when using the Distributive Property. This property is a key algebraic rule that helps to remove parentheses by distributing the multiplication over each term in the expression inside the parentheses.

Imagine you have an expression like \(7(4x - 3)\). The Distributive Property lets you rewrite this as separate multiplication operations. Here, you explicitly work out \(7 \times 4x\) and \(7 \times -3\). Breaking it down step by step can help:

• Multiply 7 by \(4x\) which results in \(28x\). This multiplication involves numbers and variables; in this case, 4 is the coefficient of \(x\).
• Next, apply 7 to \(-3\). Performing this operation results in \(-21\).

The result is an expression without parentheses: \(28x - 21\). Multiplication is thus vital to transform and prepare an expression for the next steps towards simplification.

Simplification

Once you have completed the multiplication process, the next goal is to simplify your expression. Simplification makes algebraic expressions easier to understand and work with by reducing them into their simplest form.

Taking \(28x - 21\) from the prior multiplication step, check for any further simplifications. Simplification involves:

• Combining like terms: Terms with the same variable and corresponding exponents can be added or subtracted together.
• Checking for common factors: If terms have a common factor, factor it out to simplify the expression easily.
• Ensuring the expression is in its most reduced form.

In \(28x - 21\), you don't have like terms to combine or any further factoring possible, which means our expression \(28x - 21\) is already in its simplest form.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra that contain numbers, variables, and operations. Understanding how to correctly manipulate them is crucial.

An expression like \(7(4x - 3)\) has two main parts: numerical coefficients (in this case, 7 and 4) and variable parts (represented by \(x\)).

The variable \(x\) represents an unknown value and is an essential part of solving algebraic expressions. The operations you perform, such as multiplication and simplification, help transform these expressions into an easily understandable or more useful form.

When working with algebraic expressions:

• Identify the different components: coefficients, variables, and constants.
• Use properties like the Distributive Property to multiply and simplify the expression step by step.
• Ensure each transformation you make validly follows algebraic rules to arrive at the correct final result.

Mastering these expressions allows for better problem-solving and lays down the foundation for more complex algebraic equations.