Problem 38 - 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. $$ 3(8 x-1) $$

Step-by-Step Solution

Answer

The expression simplifies to \( 24x - 3 \).

Step 1: Apply the Distributive Property

The distributive property states that \[ a(b + c) = ab + ac \] In this case, the expression is \( 3(8x - 1) \). So, we will distribute the 3 to both \( 8x \) and \( -1 \).Compute:\[ 3 \cdot 8x = 24x \]\[ 3 \cdot (-1) = -3 \]Thus, applying the distributive property gives us: \( 24x - 3 \).

Step 2: Simplify the Expression

The expression \( 24x - 3 \) is already as simplified as possible. It is in its simplest form because there are no further like terms to combine.

Key Concepts

Algebra

Algebra is like a building block of math, involving symbols and letters to represent numbers and operations. It's a bit like a puzzle, where you find the unknown. At its core, algebra helps us form mathematical expressions that can solve real-world problems. It's essential because it allows us to represent general relationships between quantities using variables.

In the exercise we're looking at, algebra plays its role by defining variables (like "x") and constants (like 3 and 8). This lets us manipulate and transform equations, whatever the value of "x" may be. This flexibility is what makes algebra so powerful.

• Variables represent unknown values.
• Constants are fixed values.
• We use operations like addition, subtraction, and multiplication to build equations.

With algebra, learning concepts like the distributive property becomes more straightforward. It's the foundation that lets us see how different numbers interact.

Simplification

Simplification in mathematics is all about making expressions more understandable and manageable. It's like cleaning up a cluttered room, leaving only the essentials behind. When we simplify an expression, we remove parentheses, combine like terms, and generally make the expression as tidy as possible.

In the original exercise, after applying the distributive property, we get a new expression: \( 24x - 3 \). This expression is already simplified because:

• There are no more parentheses to deal with.
• No like terms can be combined.

To determine if an expression is simplified, look for:

• No parentheses left to be expanded.
• All like terms combined.

In this exercise, simplification gave us the neatest form of our expression. By simplifying, we make it easier to evaluate or use in further calculations.

Mathematical Expressions

Mathematical expressions are, at their core, a way to represent numbers and operations concisely. They consist of numbers, variables, and operation symbols (like +, -, ×, and ÷). Unlike equations, expressions do not have an "equals" sign and are not statements but collections of terms.

In the given exercise, the expression starts as \( 3(8x - 1) \). This includes:

• The constant 3, which multiplies the entire expression inside the parentheses.
• A subtraction operation inside the parentheses affecting the terms \( 8x \) and 1.

The key to handling mathematical expressions is understanding how operations affect terms. The distributive property helps by allowing us to remove parentheses and produce a simpler expression.

Understanding how to interpret and transform expressions is fundamental in mathematics. It allows for deeper comprehension and the ability to tackle complex problems efficiently. With practice, mathematical expressions become intuitive and easier to work with.