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Chapter 7: Problem 21

Add or subtract. Write answer in lowest terms. \(\frac{y^{2}-3 y}{y+3}+\frac{-18}{y+3}\)

Short Answer

Expert verified
y - 6

Step by step solution

01

Identify the Common Denominator

Both fractions have the same denominator, which is \(y+3\). So, the common denominator is \(y+3\).
02

Combine the Numerators

Since the denominators are the same, combine the numerators: \( \frac{y^{2}-3y}{y+3} + \frac{-18}{y+3} = \frac{(y^{2}-3y) + (-18)}{y+3} = \frac{y^2 - 3y - 18}{y+3} \).
03

Factorize the Numerator

Factorize the quadratic expression in the numerator: \( y^2 - 3y - 18 \) factors into \( (y-6)(y+3) \), so the expression becomes \( \frac{(y-6)(y+3)}{y+3} \).
04

Simplify the Fraction

Cancel the common factor \( y+3 \) in both the numerator and the denominator: \( \frac{(y-6)(y+3)}{y+3} = y-6 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition of Rational Expressions
To add rational expressions, first look for a common denominator. This is similar to adding simple fractions like \(\frac{1}{4}+\frac{3}{4}\). Both fractions need to have the same denominator. In our problem, the denominators are already the same, which is \(y+3\). Once we have matching denominators, we can simply combine the numerators. So, \(\frac{y^{2}-3 y}{y+3}+\frac{-18}{y+3}=\frac{(y^{2}-3 y)+(-18)}{y+3}\). This step is crucial for simplifying our work later on.
Factoring Quadratics
Factoring quadratics is needed to simplify expressions. Let鈥檚 break down the quadratic expression in the numerator \(y^{2}-3y-18\). Factoring means writing it as a product of simpler binomials. We look for two numbers that multiply to \(-18\) and add up to \(-3\). Those numbers are \(-6\) and \(+3\), so we can rewrite it as \((y-6)(y+3)\). Thus, we transform \(\frac{y^{2}-3y-18}{y+3}\) into \(\frac{(y-6)(y+3)}{y+3}\). This factoring step makes the next simplification possible.
Simplifying Rational Expressions
Simplifying rational expressions involves canceling common factors. After factoring the numerator, if you see the same binomial in both the numerator and denominator, you can cancel them out. In our example, we notice \(y+3\) in both parts of the fraction \(\frac{(y-6)(y+3)}{y+3}\). So, we cancel out the \(y+3\) and are left with \(y-6\). This gives us the simplest form of the expression: \y-6\. Be careful though鈥攐nly cancel terms that are multiplied (not added or subtracted)!

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Most popular questions from this chapter

Bill Veeck was the owner of several major league baseball teams in the 1950 s and \(1960 \mathrm{s}\). He was known to often sit in the stands and en- joy games with his paying customers. Here is a quote attributed to him: I have discovered in 20 years of moving around a ballpark, that the knowledge of the game is usually in inverse proportion to the price of the seats. Explain in your own words the meaning of his statement. (To prove his point, Veeck once allowed the fans to vote on managerial decisions.)

Solve each problem involving direct or inverse variation. If \(d\) varies directly as \(t,\) and \(d=150\) when \(t=3,\) find \(d\) when \(t=5\)

The average number of vehicles waiting in line to enter a sports arena parking area is approximated by the rational expression $$\frac{x^{2}}{2(1-x)}$$ where \(x\) is a quantity between 0 and 1 known as the traffic intensity. (Source: Mannering, E., and W. Kilareski, Principles of Highway Engineering and Traffic Control, John Wiley and Sons.) To the nearest tenth, find the average number of vehicles waiting if the traffic intensity is the given number. (a) 0.1 (b) 0.8 (c) 0.9 (d) What happens to waiting time as traffic intensity increases?

Solve each variation problem. The area of a circle varies directly as the square of its radius. A circle with radius 3 in. has area 28.278 in. What is the area of a circle with radius 4.1 in. (to the nearest thousandth)?

Solve each formula for \(k\) $$ y=\frac{k}{x^{2}} $$
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