Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$\frac{1}{3}(7 x-21)$$

Step-by-Step Solution

Answer

The expression \( \frac{1}{3}(7 x-21) \) simplifies to \( \frac{7}{3}x - 7 \) using the distributive property.

Step 1: Identify the multiplicative factor for distribution

In the expression \( \frac{1}{3}(7 x-21) \), the factor that is going to be distributed is \( \frac{1}{3} \).

Step 2: Distribute the \( \frac{1}{3} \) to the terms inside the parentheses

Apply the multiplier \( \frac{1}{3} \) to both terms inside the parentheses. This yields: \( \frac{1}{3} * 7x \) and \( \frac{1}{3} * -21 \).

Step 3: Multiply the fractions

Carry out the multiplication with the fractions, yielding: \( \frac{7}{3}x - 7 \).

Key Concepts

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions is crucial for making complex problems more manageable. For instance, when we simplify an expression, we aim to make it as concise as possible. It often involves reducing fractions to their lowest terms, combining like terms, and eliminating unnecessary parentheses.

Take the expression \( \frac{1}{3}(7x-21) \) as an example. To simplify this expression, we look for common factors and use algebraic properties to our advantage. Notice that both terms within the parentheses, 7x and -21, have a common factor of 7. This commonality can be used to simplify the expression even further after applying the distributive property.

After distributing the \( \frac{1}{3} \) across the terms, we get \( \frac{7}{3}x \) and \( -7 \) which is obtained by dividing each term by the common factor of 3. The result, \( \frac{7}{3}x - 7 \) is our simplified algebraic expression. It is in its reduced form, with no parentheses, making it easier to understand and use in further calculations.

Multiplying Fractions in Algebra

When multiplying fractions in algebra, the key is to work numerically with the coefficients and symbolically with the variables. For example, in the expression \( \frac{1}{3}(7x-21) \) the first step is to distribute the fraction across the terms inside the parentheses. This involves multiplying the numerical fraction \( \frac{1}{3} \) by each term separately.

Thus, \( \frac{1}{3} \times 7x \) becomes \( \frac{7}{3}x \) or \( 2\frac{1}{3}x \) when expressed as a mixed number. Similarly, multiplying \( \frac{1}{3} \times -21 \) gives us \( -7 \) since \( \frac{1}{3} \times 21 = 7 \). The algebraic manipulation involving the fraction is straightforward—multiply the numerator (top number) by the coefficient of the variable or the constant, and keep the denominator (bottom number) unchanged. Multiplying fractions in algebra allows us to simplify our expressions for easier use in further equations or real-life applications.

Eliminating Parentheses in Algebra

Eliminating parentheses is a fundamental step in solving algebraic expressions. The distributive property is our main tool for this task. In essence, it allows us to remove the parentheses by distributing a factor outside the parentheses to each term on the inside.

For the expression \( \frac{1}{3}(7x-21) \) we use the distributive property to apply the multiplicative factor of \( \frac{1}{3} \) to 7x and -21, which are inside the parentheses. This is done by multiplying \( \frac{1}{3} \) by each term, thus integrating the factor into each term of the polynomial. By proceeding with these steps—identifying the factor, distributing it across the terms inside the parentheses, and multiplying—we have eliminated the parentheses from the expression.

This not only simplifies the expression but also prepares it for further algebraic operations, such as solving for variables. Eliminating parentheses is a skill that is repeatedly used in algebra to simplify expressions and solve equations more effectively.