In Exercises 61–66, find all values of x satisfying the given conditions. y1 = 5/(x + 4), y2 = 3/(x + 3), y3 = (12x + 19)/(x^2 + 7x + 12). and y1 + y2 = y3.

Verified step by step guidance

1. Start by setting up the equation based on the given condition: \( y_1 + y_2 = y_3 \). Substitute the expressions for \( y_1, y_2, \) and \( y_3 \) to get \( \frac{5}{x + 4} + \frac{3}{x + 3} = \frac{12x + 19}{x^2 + 7x + 12} \).
2. Find a common denominator for the left side of the equation. The common denominator for \( \frac{5}{x + 4} \) and \( \frac{3}{x + 3} \) is \((x + 4)(x + 3)\).
3. Rewrite the left side of the equation with the common denominator: \( \frac{5(x + 3) + 3(x + 4)}{(x + 4)(x + 3)} \). Simplify the numerator: \( 5(x + 3) + 3(x + 4) = 5x + 15 + 3x + 12 = 8x + 27 \).
4. Now, equate the two sides: \( \frac{8x + 27}{(x + 4)(x + 3)} = \frac{12x + 19}{x^2 + 7x + 12} \). Notice that \( x^2 + 7x + 12 \) can be factored as \((x + 4)(x + 3)\), so the denominators are the same.
5. Since the denominators are the same, set the numerators equal to each other: \( 8x + 27 = 12x + 19 \). Solve this equation for \( x \) by isolating \( x \) on one side.

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this question, y1, y2, and y3 are rational functions where the numerator and denominator are polynomials. Understanding how to manipulate and combine these functions is essential for solving the equation y1 + y2 = y3.

Finding Common Denominators

To add or equate rational functions, it is often necessary to find a common denominator. This involves identifying a common base for the denominators of the functions involved, which allows for the combination of the fractions into a single expression. This step is crucial for simplifying the equation and solving for x.

Solving Polynomial Equations

Once the rational functions are combined, the resulting equation may lead to a polynomial equation that needs to be solved for x. This involves techniques such as factoring, using the quadratic formula, or applying synthetic division. Mastery of these methods is vital for finding all possible values of x that satisfy the given conditions.