Problem 60 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{y+3}{y^{2}-11 y+10}-\frac{y+1}{y^{2}+3 y-4} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $\frac{2(8y+11)}{(y-10)(y-1)(y+4)}$.

Step 1: Factor the denominators

We need to factor the denominators of both fractions to find the LCD: $$ y^2 - 11y + 10 = (y-10)(y-1) $$ $$ y^2 + 3y - 4 = (y+4)(y-1) $$ So, the given expression is: $$ \frac{y+3}{(y-10)(y-1)} - \frac{y+1}{(y+4)(y-1)} $$

Step 2: Find the LCD

The least common denominator (LCD) is obtained by considering the unique factors in both denominators. Here, we have \((y-10)\), \((y+4)\) and \((y-1)\) as unique factors. Thus, the LCD is: $$ (y-10)(y-1)(y+4) $$

Step 3: Rewrite the fractions with the LCD and combine

Now we'll rewrite both fractions with the same denominators and combine them into a single fraction: $$ \frac{(y+3)(y+4)}{(y-10)(y-1)(y+4)} - \frac{(y+1)(y-10)}{(y+4)(y-1)(y-10)} = \frac{(y+3)(y+4) - (y+1)(y-10)}{(y-10)(y-1)(y+4)} $$

Step 4: Simplify the expression if possible

Now, let's expand the numerator and simplify: $$ \frac{y^2+7y+12 - (y^2 - 9y - 10)}{(y-10)(y-1)(y+4)} = \frac{y^2+7y+12-y^2+9y+10}{(y-10)(y-1)(y+4)} $$ Combine the terms in the numerator: $$ \frac{16y+22}{(y-10)(y-1)(y+4)} $$ Now we simplify the numerator: $$ \frac{2(8y+11)}{(y-10)(y-1)(y+4)} $$ So, the final simplified expression is: $$ \frac{2(8y+11)}{(y-10)(y-1)(y+4)} $$

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is an essential tool in simplifying rational expressions and solving algebraic equations. Quadratics often appear in the form \( ax^2 + bx + c \). The first step in subtracting rational expressions is to factor these quadratic expressions in the denominators.

In the given exercise, we start with two quadratic expressions in the denominators:

• For \( y^2 - 11y + 10 \), we seek two numbers that multiply to 10 (the constant term) and sum to -11 (the coefficient of the middle term). These numbers are -1 and -10, so it factors into \((y-10)(y-1)\).
• Similarly, for \( y^2 + 3y - 4 \), we need two numbers that multiply to -4 and add to 3. These numbers are 4 and -1, thus it factors into \((y+4)(y-1)\).

Notice that factoring quadratics often involves trial and error, but with practice, recognizing patterns becomes easier. This kind of factoring is particularly useful for finding a common denominator.

Least Common Denominator

The least common denominator (LCD) is the smallest denominator that can accommodate all fractions involved, which makes combining expressions possible. When rational expressions have different denominators, it's crucial to convert them to a common denominator to perform operations on them.

In our example, once we've factored the denominators, the LCD is determined by identifying all unique factors from each expression’s denominator:

• From the first fraction’s denominator \((y-10)(y-1)\), we list \( (y-10) \) and \( (y-1) \).
• From the second fraction’s denominator \((y+4)(y-1)\), we add \( (y+4) \).
• The factor \( (y-1) \) is present in both denominators, so it only needs to be included once in the LCD.

Thus, the least common denominator becomes \( (y-10)(y-1)(y+4) \). With this common base, each fraction is rewritten to share the same denominator, allowing for straightforward subtraction.

Simplifying Expressions

Finally, simplifying expressions is the process of reducing an algebraic expression to its simplest form. This involves combining like terms, factoring, and reducing any unnecessary terms.

In the exercise, after we find the LCD and rewrite both fractions, we then combine them into a single rational expression. This requires distributing and combining the terms in the numerator of the expression:

• The expression \( (y+3)(y+4) - (y+1)(y-10) \) needs to be expanded in the numerator.
• After expanding, you will arrive at \( y^2 + 7y + 12 - (y^2 - 9y - 10) \).
• Combine like terms: the \( y^2 \) terms cancel out, leaving us with \( 16y + 22 \) in the numerator.

Further simplification transforms \( 16y + 22 \) into \( 2(8y+11) \), after factoring out the greatest common factor. The ultimate simplified form of the expression results in \( \frac{2(8y+11)}{(y-10)(y-1)(y+4)} \), which is the simplest representation of the operation performed.