Problem 10 - Algebra 1

Question

Use the distributive property to rewrite the expression without parentheses. $$ -\frac{2}{3}(t-24) $$

Step-by-Step Solution

Answer

The expression \(-\frac{2}{3}(t-24)\) without parentheses is \(-\frac{2}{3}t + 16\).

Step 1: Identify the distributive property

The distributive property of multiplication over subtraction is: \(a * (b - c) = a * b - a * c\). Here \(a = -\frac{2}{3}\), \(b = t\) and \(c = 24\). The task is to apply this property to the given expression.

Step 2: Apply the distributive property

Apply the distributive property: \(-\frac{2}{3} * t - (-\frac{2}{3} * 24)\). This involves multiplying -2/3 with 't' and -2/3 with '24' separately.

Step 3: Simplify the expression

After multiplication, the expression becomes: \(-\frac{2}{3}t + 16\). The negative sign before 2/3 has changed to positive when multiplied with 24 because the multiplication of two negative numbers gives a positive result.

Key Concepts

Multiplication Over Subtraction

In mathematics, the distributive property is a powerful tool that allows us to simplify expressions involving subtraction and multiplication. The key idea is that multiplication impacts each part of the subtraction separately. This property is expressed mathematically as:

• \(a \times (b - c) = a \times b - a \times c\)

For example, if we take the expression \(-\frac{2}{3}(t-24)\), the distributive property helps us break down the operation:

• First, multiply \(-\frac{2}{3}\) by \(t\)
• Then, multiply \(-\frac{2}{3}\) by \(-24\)

Notice how each term inside the parentheses is individually multiplied by \(-\frac{2}{3}\). This technique is essential for solving algebraic equations and simplifying complex problems.

Simplifying Expressions

Simplifying expressions is a necessary process in algebra that involves rewriting expressions in a more concise and manageable form. After applying the distributive property, the goal is to clean up the expression, which often involves combining like terms or reducing coefficients. In our original problem, after distributing \(-\frac{2}{3}\):

• First term: \(-\frac{2}{3} \times t = -\frac{2}{3}t\)
• Second term: \(-\frac{2}{3} \times (-24) = +16\)

The simplification step clarifies that multiplying two negatives yields a positive. By rewriting the expression as \(-\frac{2}{3}t + 16\), we make it easier to understand and use in further calculations. Simplifying expressions is key to solving equations efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent specific values or relationships. They can be as simple as \(x + 2\) or more complex like our exercise: \(-\frac{2}{3}(t - 24)\).Understanding algebraic expressions involves recognizing:

• Variables such as \(t\) that stand for unknown values
• Constants such as \(-\frac{2}{3}\) that are fixed values
• Operations like addition and multiplication that connect the parts

Manipulating algebraic expressions, especially using properties like distribution, enables us to simplify and solve them. These skills are fundamental in math, allowing us to handle real-world problems and create mathematical models. The exercise demonstrates the importance of applying properties systematically to extract meaningful and simplified expressions.