Problem 55 - Prealgebra

Question

Apply the distributive property, then simplify. $$\frac{1}{2}(12 x+6)$$

Step-by-Step Solution

Answer

The simplified expression is \( 6x + 3 \).

Step 1: Understand the Distributive Property

The distributive property states that multiplying a single term outside the parentheses by each term within the parentheses, and then adding those results, gives the same result as simplifying inside the parentheses first and multiplying the resulting sum. It ensures that every term inside the parentheses is multiplied by the factor outside the parentheses.

Step 2: Apply the Distributive Property

Using the distributive property, we multiply each term inside the parentheses by the factor outside the parentheses. In the expression \( \frac{1}{2}(12x + 6) \), distribute \( \frac{1}{2} \) over both \( 12x \) and \( 6 \). This multiplication is done as follows: \[ \frac{1}{2} \times 12x + \frac{1}{2} \times 6 \]

Step 3: Perform the Multiplications

Compute the products obtained during the distribution:- \( \frac{1}{2} \times 12x = 6x \)- \( \frac{1}{2} \times 6 = 3 \)Thus, the expression becomes \( 6x + 3 \).

Step 4: Simplify the Expression

After distributing and performing the multiplications, the expression is already in its simplest form, resulting in \( 6x + 3 \). There are no like terms to combine, so no further simplification is necessary.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in algebra. It involves manipulating an expression to make it easier to work with. This often means condensing the expression into its simplest form.
To simplify, follow these basic steps:

• Distribute any terms: The distributive property allows us to multiply each term inside a set of parentheses by a factor that is outside the parentheses.
• Combine like terms: Terms that have the same variable and the same exponent can be combined by adding or subtracting their coefficients.
• Simplify fractions and constants: Finally, ensure any fractions or constants are in their simplest form.

This systematic approach ensures the expression is in a form that is both straightforward and functional for further calculations.

Multiplying Fractions

Multiplying fractions is a fundamental operation in algebra. When you multiply fractions, simply multiply the numerators together and the denominators together.
For example, consider multiplying the fractions \( \frac{a}{b} \times \frac{c}{d} \). This results in:

• Numerator: \( a \times c \)
• Denominator: \( b \times d \)

Thus, \( \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \).
This straightforward method makes calculations involving fractions less daunting. Remember, always simplify your final fraction when possible to make your answers cleaner and more understandable.

Basic Algebra Concepts

Understanding basic algebra concepts is crucial for progressing in math. This includes familiarity with variables, constants, expressions, and equations.
Key points include:

• Variables: Symbols (like \( x \) or \( y \)) used to represent numbers whose values are not yet known.
• Constants: Numbers within an expression that do not change their value.
• Expressions: Combinations of variables and constants that represent a mathematical reality.
• Equations: Mathematical statements that assert the equality of two expressions.

Grasping these fundamentals is vital. They form the building blocks for more complex mathematical operations and problem-solving down the line.