Problem 55 - Algebra and Trigonometry

Question

Find all values of x satisfying the given conditions. $$\begin{aligned}&y_{1}=\frac{5}{x+4}, y_{2}=\frac{3}{x+3}, y_{3}=\frac{12 x+19}{x^{2}+7 x+12}, \text { and }\\\&y_{1}+y_{2}=y_{3}\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the problem is x = 2

Step 1: Combine \(y_{1}\) and \(y_{2}\) into a single fraction

Since the denominators only differ by 1, they can be made the same by multiplying \(y_{1}\) by \((x + 3)/(x + 3)\) and adding it to \(y_{2}\)\nThis results in: \(y_{1}+y_{2} = \frac{5(x + 3) + 3(x + 4)}{(x + 4)(x + 3)} = \frac{8x + 27}{x^{2} + 7x + 12}\)

Step 2: Set the sum equal to \(y_{3}\)

The equation is now: \(\frac{8x + 27}{x^{2} + 7x + 12} = \frac{12x + 19}{x^{2} + 7x + 12}\)

Step 3: Simplify

Since the denominators of both sides of the equation are the same, they can be subtracted from both sides, leading to the equation: \(8x + 27 = 12x + 19\)

Step 4: Solve for x

Subtracting 8x from both side gives: 4x + 19 = 27\nSubtracting 19 from both sides gives: 4x = 8\nDividing by 4 gives the final solution: x = 2

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but contain variables in the numerator, the denominator, or both. In the given problem, we encountered algebraic fractions like \( y_1 = \frac{5}{x+4} \), showcasing a variable in the denominator. Understanding how to handle these expressions is vital for manipulating and solving equations involving fractions.

Equation Solving

To solve equations with algebraic fractions, we follow a systematic approach. First, analyze the denominators and find a common denominator to combine terms, as seen in the original problem where \( y_1 + y_2 \) was combined. With common denominators, the fractions can be simplified to a single term, and the numerator of the resulting fraction is then set equal to the numerator of the fraction on the other side of the equation. Once the fractions are eliminated, we employ classic techniques such as isolating the variable on one side of the equation to find the solution.

Fraction Addition and Simplification

When adding algebraic fractions, it's essential to find a common denominator, as it allows you to sum the numerators while keeping the denominator the same. Like in the provided exercise, where \( y_1 \) was multiplied by \( (x+3)/(x+3) \) to align it with \( y_2 \) which shares the similar denominator with \( y_3 \). This process, combined with simplifying the resulting fraction, ultimately leads to a simpler equation without fractions, making it easier to solve for the unknown variable. Always ensure the fractions are fully simplified before proceeding to the next step to avoid any algebraic errors.