Problem 76 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -\frac{1}{3}(9 x+5) $$

Step-by-Step Solution

Answer

-3x - \frac{5}{3}

Step 1 - Identify the Distributive Property

The distributive property states that for any numbers a, b, and c: \[ a(b + c) = ab + ac \]In this case, the expression \[ -\frac{1}{3}(9x + 5) \] can be rewritten by distributing the \[ -\frac{1}{3} \] to both terms inside the parenthesis.

Step 2 - Distribute to the First Term

Multiply \[ -\frac{1}{3} \] by \[ 9x \]:\[ -\frac{1}{3} \times 9x = -3x \]

Step 3 - Distribute to the Second Term

Multiply \[ -\frac{1}{3} \] by \[ 5 \]:\[ -\frac{1}{3} \times 5 = -\frac{5}{3} \]

Step 4 - Combine the Results

Put both parts together to get the final expression:\[ -3x - \frac{5}{3} \]

Key Concepts

distributive property

Understanding the distributive property is key to solving many algebra problems. The distributive property helps us to simplify multiplication of a single term across a sum or difference inside parentheses. To put it simply, when you have an expression like \(a(b + c)\), you distribute \(a\) to both \(b\) and \(c\).
This results in the equation \(ab + ac\).

In our exercise, we applied the distributive property to the expression \(-\frac{1}{3}(9x + 5)\).
We multiplied \(-\frac{1}{3}\) by each term inside the parentheses separately:

• First, \( -\frac{1}{3} \times 9x = -3x\)
• Then, \( -\frac{1}{3} \times 5 = -\frac{5}{3} \)

This results in the simplified expression \(-3x - \frac{5}{3}\). Using the distributive property not only simplifies expressions but also prepares us for more complex algebraic manipulations.

algebraic expressions

An algebraic expression is a mix of numbers, variables, and arithmetic operations. In the given problem, we started with the algebraic expression \(-\frac{1}{3}(9x + 5)\).

This expression includes:

• A numerical fraction: \(-\frac{1}{3}\)
• A variable term: \(9x\)
• A constant term: 5

The goal was to simplify this expression using algebraic rules, such as the distributive property. By applying the distributive property, we broke it down into simpler parts:

• The variable term became \(-3x\)
• The constant term turned into \(-\frac{5}{3}\)

Combining these simplified parts gives us the final expression \(-3x - \frac{5}{3}\). This shows how algebraic expressions can be manipulated and simplified for easier handling.

multiplication of fractions

Multiplying fractions might seem tricky, but it's very straightforward once you get the hang of it. When you multiply fractions, you multiply the numerators together and the denominators together. For example, in our exercise, one of the steps involved multiplying the fraction \(-\frac{1}{3}\) by 9.
To do this, consider 9 as \(\frac{9}{1}\), so the multiplication looks like:

\ \ $$ -\frac{1}{3} \times \frac{9}{1} = -\frac{1 \times 9}{3 \times 1} = -\frac{9}{3} = -3 $$

We also multiplied \(-\frac{1}{3}\) by 5:
Consider 5 as \(\frac{5}{1}\), so the multiplication goes like this:

• \(-\frac{1}{3} \times \frac{5}{1} = -\frac{1 \times 5}{3 \times 1} = -\frac{5}{3}\)

Breaking down the multiplication of fractions into these simple steps makes it easier to understand and apply in algebraic problems.