Step-by-step Solution:

Let the equation of the line be: \[ \frac{x}{a} + \frac{y}{b} = 1 \] where \(a\) and \(b\) are the x-intercept and y-intercept, respectively. Given that the sum of the intercepts is 12: \[ a + b = 12 \quad \Rightarrow \quad b = 12 - a \] Since the line passes through the point \((2, 3)\), substituting into the equation: \[ \frac{2}{a} + \frac{3}{b} = 1 \] Substitute \(b = 12 - a\): \[ \frac{2}{a} + \frac{3}{12 - a} = 1 \] Multiply through by \(a(12 - a)\) to eliminate the denominators: \[ 2(12 - a) + 3a = a(12 - a) \] \[ 24 - 2a + 3a = 12a - a^2 \] \[ 24 + a = 12a - a^2 \] \[ a^2 - 11a + 24 = 0 \] Solve the quadratic equation: \[ a = \frac{11 \pm \sqrt{121 - 96}}{2} = \frac{11 \pm 5}{2} \] \[ a = 8 \quad \text{or} \quad a = 3 \] For \(a = 8\): \[ b = 12 - 8 = 4 \] Equation of the line: \[ \frac{x}{8} + \frac{y}{4} = 1 \quad \Rightarrow \quad x + 2y = 8 \] For \(a = 3\): \[ b = 12 - 3 = 9 \] Equation of the line: \[ \frac{x}{3} + \frac{y}{9} = 1 \quad \Rightarrow \quad 3x + y = 9 \] Now, check which of the given options match these equations: - (A) \(3x + y = 9\) (matches) - (B) \(x + 3y = 9\) (does not match) - (C) \(x + 2y = 8\) (matches) - (D) \(5x + 7y = 35\) (does not match) Correct Equations: - (A) and (C) Final Answer: \[ \boxed{C} \]