- Identify the common denominator as \((w-2)(w-3) = w^2 - 5w + 6\).
- Rewrite each fraction using this denominator: \(\frac{4w(w-3)}{w^2 - 5w + 6} + \frac{3w(w-2)}{w^2 - 5w + 6}\).
- Simplify the sum by combining the numerators: \(\frac{4w^2 - 12w + 3w^2 - 6w}{w^2 - 5w + 6} = \frac{7w^2 - 18w}{w^2 - 5w + 6}\).
- The correct option is \(\boxed{\frac{7w^2 - 18w}{w^2 - 5w + 6}}\).
Explanation
- Finding a Common Denominator
We are tasked with simplifying the expression \(\frac{4w}{w-2} + \frac{3w}{w-3}\). To add two rational expressions, we must first find a common denominator. The denominators are \((w-2)\) and \((w-3)\). The least common denominator is their product: \((w-2)(w-3)\).Expanding the denominator gives: \((w-2)(w-3) = w^2 - 3w - 2w + 6 = w^2 - 5w + 6\).
- Rewriting Fractions with the Common Denominator
To combine the fractions, we rewrite each one with the common denominator \((w^2 - 5w + 6)\):
\(\frac{4w}{w-2} = \frac{4w(w-3)}{(w-2)(w-3)} = \frac{4w^2 - 12w}{w^2 - 5w + 6}\)
\(\frac{3w}{w-3} = \frac{3w(w-2)}{(w-3)(w-2)} = \frac{3w^2 - 6w}{w^2 - 5w + 6}\)
- Adding the Numerators and Concluding
Now that the fractions share the same denominator, we can add the numerators:
\(\frac{4w^2 - 12w}{w^2 - 5w + 6} + \frac{3w^2 - 6w}{w^2 - 5w + 6} = \frac{(4w^2 - 12w) + (3w^2 - 6w)}{w^2 - 5w + 6}\)
Combining like terms in the numerator:
\((4w^2 + 3w^2) + (-12w - 6w) = 7w^2 - 18w\).
Thus, the expression simplifies to \(\frac{7w^2 - 18w}{w^2 - 5w + 6}\). This matches the first given option.
Examples
Rational expressions are used in various fields such as engineering and physics, particularly in circuit analysis or calculating combined rates. For example, if two workers can complete a task in \((w-2)\) and \((w-3)\) hours respectively, the combined rate of work, which is the sum of their individual rates, can be represented as a rational expression. Simplification helps in understanding the total time required or the efficiency of the system, making it easier to solve for specific values of \(w\).