Problem 57 - Intermediate Algebra

Question

Multiply. See Example 4 $$3\left(\frac{4}{3} x-\frac{5}{3} y+\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(4x - 5y + 1\).

Step 1: Distribute the Constant Across the Terms

First, distribute the 3 to each term inside the parentheses. This means you'll multiply 3 by each fraction:\[3 \times \left(\frac{4}{3} x\right),\ 3 \times \left(-\frac{5}{3} y\right),\ 3 \times \left(\frac{1}{3}\right)\].

Step 2: Simplify Each Fraction Multiplication

For each term, multiply the numerator by 3 and simplify the fraction:\(3 \times \frac{4}{3} x = 4x\),\(3 \times -\frac{5}{3} y = -5y\),\(3 \times \frac{1}{3} = 1\).

Step 3: Rewrite the Expression in Simplified Form

Combining the results from Step 2 gives the simplified expression:\[4x - 5y + 1\]

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental principle in algebra that simplifies the multiplication of numbers and variables. It's particularly useful when dealing with expressions inside parentheses. This property states that multiplying a single term by terms inside parentheses is the same as doing the multiplication separately and then adding the results. In mathematical terms, if you have a scenario like \(a(b + c)\), it is the same as \(ab + ac\).

For example, in the given problem, the expression \(3\left(\frac{4}{3} x - \frac{5}{3} y + \frac{1}{3}\right)\) requires the application of the distributive property. Here, "3" is distributed to each term inside the parentheses which simplifies the expression step-by-step.

The key takeaway is recognizing the efficiency and simplicity this property brings when working with algebraic expressions, allowing us to break down complex problems into easier components.

Mastering Fraction Multiplication

Fraction multiplication can initially seem tricky, but it becomes straightforward once you understand the basic rules. Multiplying fractions involves multiplying the numerators across and the denominators across. The rule is simple: if you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), their product is \(\frac{ac}{bd}\).

In our specific problem, when multiplying each term by "3," it effectively simplifies the process as follows:

• Multiply the numerator "3" by the numerators of the fractions inside the parentheses,
• Then, simplify wherever possible.
  

For example, in \(3 \times \frac{4}{3}x\), the "3" in the numerator and denominator cancel each other, leaving us with just "4x." This same method applies to \(-\frac{5}{3}y\) and \(\frac{1}{3}\). Recognizing cancellation opportunities makes fractional multiplication much simpler and results in an elegant simplification process.

Simplifying Expressions Effectively

Simplifying algebraic expressions involves reducing them to their simplest form. This process includes eliminating any unnecessary parentheses, combining like terms, and simplifying any fractions if possible.

In the exercise at hand, after distributing and multiplying, what remains is the combination of like terms where applicable and recognizing that the expression in its simplest form is succinct and clear. The expression simplifies to \(4x - 5y + 1\).

Simplification helps to make expressions more comprehensible and ready for further algebraic manipulation or evaluation. It's always about making the algebraic expressions as clear and as efficient as possible, aiding both in the understanding and solving of complex mathematical problems.