Problem 13 - Algebra 2

Question

Simplify each expression. $$ \frac{1}{2}(16-4 a)-\frac{3}{4}(12+20 a) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-1 - 17a\).

Step 1: Distribute the First Term

Apply the distributive property to the first term. Multiply \( \frac{1}{2} \) by each term inside the parentheses: \( \frac{1}{2}(16) - \frac{1}{2}(4a) = 8 - 2a \).

Step 2: Distribute the Second Term

Apply the distributive property to the second term. Multiply \( -\frac{3}{4} \) by each term inside the parentheses: \( -\frac{3}{4}(12) - \frac{3}{4}(20a) = -9 - 15a \).

Step 3: Combine Like Terms

Combine the results from Steps 1 and 2: \( (8 - 2a) + (-9 - 15a) \). Now combine like terms: \( 8 - 9 = -1 \) and \( -2a - 15a = -17a \). Thus, the expression simplifies to \(-1 - 17a \).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps to simplify expressions by removing parentheses. This rule states that a term outside the parentheses can be multiplied by each term inside the parentheses. For example, given an expression like \(a(b + c)\), you can simplify it as \(ab + ac\). It allows you to "distribute" the term across the sums or differences within the parentheses.

In the original exercise, the goal was to simplify the expression \(\frac{1}{2}(16-4a) - \frac{3}{4}(12+20a)\).

• First, the distributive property was used to expand \(\frac{1}{2}(16-4a)\). This was done by multiplying \(\frac{1}{2}\) by both \(16\) and \(-4a\), resulting in the expression \(8 - 2a\).
• Next, the same rule applied to \(-\frac{3}{4}(12+20a)\). Each term inside the parentheses \(12\) and \(20a\) was multiplied by \(-\frac{3}{4}\), resulting in \(-9 - 15a\).

Using the distributive property helps break down complex algebraic expressions into simpler parts, making it easier to perform further operations like combining like terms.

Combining Like Terms

After applying the distributive property, the next step in simplifying an expression is often combining like terms. Like terms are terms that contain the same variables raised to the same power. Only the coefficients (numbers in front of these variables) are different.

For example, in the expression obtained from the previous step \((8 - 2a) + (-9 - 15a)\), like terms were identified:

• The constants \(8\) and \(-9\) are like terms because they both lack variables.
• The terms \(-2a\) and \(-15a\) are like terms because they both contain the same variable \(a\).

To combine like terms:

• Subtract the constant terms: \(8 - 9 = -1\).
• Combine the coefficients of terms with the same variable: \(-2a - 15a = -17a\).

Combining like terms simplifies the expression by merging similar elements together, reducing the overall complexity and preparing it for further operations or interpretations.

Simplification Process

The simplification process in algebra involves a sequence of steps to transform a given expression into its simplest form. It's about making an expression easier to understand or solve and involves a few essential operations.

For the expression \(\frac{1}{2}(16-4a) - \frac{3}{4}(12+20a)\), simplification started by expanding expressions using the distributive property, then moved onto combining like terms.

• Step 1: Use the distributive property for multiplication over addition/subtraction.
• Step 2: Combine like terms by adding or subtracting the coefficients of terms with the same variable.

As each operation is performed, the expression becomes less complicated:

• The original expression was turned into \(8 - 2a - 9 - 15a\).
• Upon combining like terms, it became \(-1 - 17a\).

This final expression, \(-1 - 17a\), is much simpler and ready for any further use you might need, such as solving an equation or graphing a function. Simplification reduces clutter and focuses your efforts on the essential elements of a mathematical problem.