Problem 64 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y^{2}-6}{y^{2}+9 y+18}-\frac{y-4}{y+6}$$

Step-by-Step Solution

Answer

The simplified result of the given expression is \(1/(y+6)\).

Step 1: Factorising the denominators

Factorise the denominators of the two expressions. The first denominator quadratic equation \(y^2+9y+18 = (y+6)(y+3)\) and the second denominator is \(y+6\). Both expressions share a common factor, \(y+6\), but the first expression has an additional factor, \(y+3\).

Step 2: Making the denominators of the two expressions the same

The second expression needs to have the same denominator as the first to carry out the subtraction. Multiply the second expression by \((y+3)/(y+3)\) which is equivalent to 1 and does not affect the overall value of the expression. Now, the two expressions are \( (y^2-6)/((y+6)(y+3)) \) and \( (y-4)(y+3)/((y+6)(y+3)) \).

Step 3: Subtracting the two expressions

Now that the denominators of both expressions are the same, the two expressions can be subtracted: \[ \frac{y^2-6-(y-4)(y+3)}{(y+6)(y+3)} \]

Step 4: Simplifying the expression

Begin by expanding and simplifying the numerator: \( y^2 - 6 - (y^2 -y -12) \) which simplifies to \( 2y + 6 \). Hence the result of the subtraction is: \[ \frac{2y + 6}{(y+6)(y+3)} \]. We can still divide the numerator and denominator by 2 further simplifying the expression to: \[ \frac{y+3}{y+3)(y+6)} \]

Step 5: Cancelling common factors

The factors \(y+3\) in the numerator and denominator cancel out, thus simplifying the expression to \(1/(y+6)\).