Problem 34 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{y+1}+\frac{2}{3 y}$$

Step-by-Step Solution

Answer

The simplified result of the subtraction operation \(\frac{y^{2}+3y}{y^{2}+y-12} - \frac{y^{2}-12}{y^{2}+y-12}\) is \(\frac{3y+12}{y^{2}+y-12}\).

Step 1: Identify Like Denominators

Both fractions have the same denominator, \(y^{2}+y-12\). This means we can subtract the fractions directly.

Step 2: Subtract Fractions

Subtract the numerators of the fractions: \(y^{2}+3y - (y^{2}-12)\). Be careful with the negative sign. You need to distribute it over the \(y^{2}-12\), this will result in \(y^{2}+3y - y^{2}+12\).

Step 3: Simplify Numerator

Simplify the result from the previous step into the following: \(3y + 12\). The \(y^{2}\) and \(-y^{2}\) cancel each other.

Step 4: Write the Final Answer

Now that the numerator has been simplified, write the answer over the original denominator. Hence, the final simplified fraction is \(\frac{3y+12}{y^{2}+y-12}\).

Key Concepts

Like Denominators in Algebra

Understanding like denominators is crucial for simplifying algebraic fractions. Like denominators occur when two or more fractions share the same bottom number, or denominator, which signifies the total number of parts the whole is divided into. In algebra, when dealing with fractions that have polynomials in their denominators, like denominators mean that the algebraic expressions are identical. For example, in the exercise \(\frac{y^{2}+3y}{y^{2}+y-12} - \frac{y^{2}-12}{y^{2}+y-12}\), both fractions have the same denominator of \(y^{2}+y-12\).

This is beneficial because it simplifies the process of addition or subtraction; there's no need for the extra step of finding a common denominator, which can be time-consuming and complex with polynomials. Having like denominators means we can move straight into combining the numerators directly, while the denominator remains untouched throughout the operation.

Subtracting Polynomials

When working with algebraic expressions such as subtracting polynomials, it is important to understand the distribution of the negative sign across the terms of the polynomial. In the exercise, we see \(y^{2}+3y - (y^{2}-12)\). Here, the negative sign in front of the parenthesis must be distributed to both terms inside, effectively changing their signs. Thus, \(y^{2} - y^{2}\) will cancel out, and \(3y + (-(-12))\) becomes \(3y + 12\).

Subtracting polynomials is much like combining like terms. You only combine the terms with the same variables raised to the same power. If a term doesn't have a like term to combine with, it simply carries down unchanged. Remembering the basic rule 'like terms combine, unlike terms don't' and being mindful to distribute the negative sign correctly will ensure that you subtract polynomials correctly and simplify to the most reduced form.

Simplifying Expressions

The final step in many algebra problems, including our textbook exercise, involves simplifying expressions. This process is crucial for finding the most compact form of an algebraic expression. Simplifying might include combining like terms, factoring, canceling terms, and/or reducing fractions. In our given subtraction problem, once we've correctly distributed our negative sign and subtracted our polynomials, we can simply combine like terms (if any) for the simplified numerator which in this case is \(3y + 12\).

This numerator does not have any like terms, so it is already in its simplest form. In other cases, you may also need to factor out the greatest common factor or potentially reduce the fraction if the numerator and denominator have common factors. Simplifying makes the expression easier to understand and work with in future operations. Regular practice is key to mastering this concept, just as with the previous concepts of like denominators and subtracting polynomials.