Step 1: Understanding the Concept:
The intercept form of a line is \( x/a + y/b = 1 \). We are given that the sum of intercepts \( a+b = 4 \).
Step 2: Detailed Explanation:
Let the intercepts be \( a \) and \( b \). Given \( a + b = 4 \implies b = 4 - a \).
Equation of line: \( \frac{x}{a} + \frac{y}{4-a} = 1 \).
The line passes through \( (9/2, -5) \):
\[ \frac{9}{2a} + \frac{-5}{4-a} = 1 \]
\[ \frac{9(4-a) - 10a}{2a(4-a)} = 1 \]
\[ 36 - 9a - 10a = 8a - 2a^2 \]
\[ 2a^2 - 27a + 36 = 0 \]
Factoring the quadratic:
\[ (2a - 3)(a - 12) = 0 \]
This gives \( a = 3/2 \) or \( a = 12 \).
Since the intercepts must be positive and their sum is 4, \( a = 12 \) is impossible because then \( b = 4 - 12 = -8 \).
Thus, \( a = 3/2 = 1.5 \).
Then \( b = 4 - 1.5 = 2.5 = 5/2 \).
Substituting into the equation:
\[ \frac{x}{3/2} + \frac{y}{5/2} = 1 \implies \frac{2x}{3} + \frac{2y}{5} = 1 \]
Multiply by 15:
\[ 10x + 6y = 15 \].
Step 3: Final Answer:
The equation is \( 10x + 6y = 15 \).