Problem 30 - Algebra 1: Concepts and Skills

Question

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{y^{2}-2 y}{y^{2}-7 y-18}-\frac{9(y-2)}{y^{2}-7 y-18} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{y - 2}{y + 2}\).

Step 1: Factorising the denominator

Firstly, factor the common denominator \(y^{2}-7y-18\). The expression can be written as \((y - 9)(y + 2)\).

Step 2: Simplifying the numerator

Now, simplify the numerator of the second fraction: \(9(y - 2)\), which equals \(9y - 18\). So the expression will look like this: \(\frac{y^{2}-2 y}{(y - 9)(y + 2)} - \frac{9y - 18}{(y - 9)(y + 2)}\).

Step 3: Combining the fractions

Since the fractions have the same denominator, they can be combined as follows: \(\frac{y^{2}-2y - (9y - 18)}{(y - 9)(y + 2)}\). This simplifies to: \(\frac{y^{2}-11y+18}{(y - 9)(y + 2)}\).

Step 4: Simplifying the fraction further

Notice that the numerator can be factored into \((y - 9)(y - 2)\). So the final simplified answer becomes: \(\frac{(y - 9)(y - 2)}{(y - 9)(y + 2)}\).

Step 5: Cancel out common term

The \(y - 9\) term in the numerator and denominator can be canceled out, resulting to \(\frac{y - 2}{y + 2}\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a foundational concept in algebra that involves reducing expressions to their simplest form. This process can make complex algebraic statements more understandable and easier to work with. When dealing with expressions like \(\frac{y^{2}-2y}{y^{2}-7y-18}-\frac{9(y-2)}{y^{2}-7y-18}\), the first step is identifying common factors and terms within the expression.

During simplification, you might find it beneficial to factor denominators and numerators, combine fractions with a common denominator, and cancel out common terms across the numerator and denominator. The aim is to present the expression in its least complicated form without changing its value. This not only makes the expressions more manageable during further operations but also helps in visualizing the underlying structure of the algebraic expression.

Combining Like Terms

Combining like terms is another essential concept in algebra which involves merging terms within an expression that have identical variable parts. Terms that are constants or have exactly the same variable(s) to the same power are considered 'like' and can be added or subtracted from each other. In the expression \(\frac{y^{2}-11y+18}{(y - 9)(y + 2)}\), the like terms within the numerator have been combined.

Understanding how to identify and combine like terms is critical for simplifying algebraic expressions. It's an efficient way to shrink the size of an equation or expression, making subsequent calculations far less complex. When dealing with polynomials, especially, the ability to effectively combine like terms is a stepping-stone toward mastery in algebra.

Factorisation of Quadratics

Understanding The Process

The process of factorizing quadratics is a method used to break down a quadratic expression into a product of two binomials. In essence, it's the reverse of multiplying out two binomials. For instance, if you have the quadratic \(y^{2}-7y-18\), factorization turns it into \(y - 9)(y + 2)\).

Why Factorize Quadratics?

Factorizing is a crucial skill because it is often the first step in solving quadratic equations, which are prevalent in various mathematical problems. It is also essential in simplifying expressions and can aid in graphing parabolas since it reveals the roots of the quadratic.

When factorizing, you're looking for two numbers that multiply to give the constant term (in this case, -18) and add to give the coefficient of the middle term (in this case, -7). Mastery in factorizing quadratics implies being able to use several techniques such as the 'ac method', completing the square, or using the quadratic formula when factors are not readily apparent.