Problem 18 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{5 x y-3 y}{9-15 x}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{y(5x - 3)}{9 - 15x}\)

Step 1: Simplify the numerator

In the numerator, factor out the common factor of \(y\) from the terms. Thus \(5xy - 3y\) simplifies to \(y(5x-3)\).

Step 2: Simplify the denominator

The denominator does not have a common variable to factor out. So, \(9 - 15x\) remain as is.

Step 3: Placement

Place the simplified numerator over the simplified denominator to obtain the final answer.

Key Concepts

Factoring Algebraic Expressions

Factoring is an essential skill when working with algebraic expressions. It involves breaking down a complex expression into simpler parts that, when multiplied together, give back the original expression. This is akin to finding what ingredients went into a cake by tasting it; you're trying to figure out the simple flavors (or expressions) that make up the whole.
To factor an algebraic expression, you look for common factors in each term. These common factors could be numbers, variables, or a combination of both. In our exercise, the numerator is factored by identifying 'y' as the common factor in both terms, which leads to the expression being rewritten as:
\[ y(5x - 3) \].
This simplification is a neat trick because it can reveal some magical cancellations or simplifications with the denominator, although it's not always guaranteed. Remember, spotting these common factors is key, just like spotting a familiar face in a crowded room. And just like in solving a puzzle, once you identify and extract these common pieces, the big picture starts to become clearer.

Simplifying Numerators and Denominators

Imagine the numerator and the denominator of a fraction as two opposing teams in a tug-of-war match. Simplifying them is like reducing the number of players to the very essential ones, ensuring that the tug-of-war is as fair as possible. In an algebraic fraction, simplifying the numerator and the denominator allows us to reduce the expression to its simplest form.
To simplify the numerator, as done with the term \( 5xy - 3y \), we looked for a common variable or number that can be factored out. Similarly, the denominator needs to be looked at with the same analytical eye. However, in some cases, you can't simplify the denominator further, as with the term \( 9 - 15x \) in our exercise. When that happens, it's like a stalemate in the tug-of-war—neither side moves.
After simplifying both the numerator and the denominator independently, the next big question is whether they can simplify each other. This isn't possible in our exercise, but it's something to watch out for because when it happens, it's like one team suddenly letting go of the rope—the simplification is swift and dramatic.

Algebraic Fraction Simplification

Once we have factored our algebraic expressions and simplified the individual parts, we can look at the fraction as a whole to see if there are any final touches we can make, like adding a dash of salt to a finished dish. Algebraic fraction simplification involves combining the previous steps to achieve the simplest form of an expression. In some cases, you can cancel out common factors from the numerator and the denominator, significantly reducing the complexity of the expression.
In our exercise, the simplified fraction is \( \frac{y(5x-3)}{9-15x} \). Here, there are no common factors between the numerator and denominator, so the tug-of-war ends in a draw. This is our final form, like the last brushstroke on a painting. Always remember, the goal of simplification is not always to make the expression short but to make it as clear and manageable as possible. In real-life problem-solving, these skills allow you to tackle more complex problems with confidence, reducing intimidation and unlocking potential solutions.