A straight line makes on the coordinate axes positive intercepts whose sum is 7. If the line passes through the point (-3, 8), find its equation.

Let the line make intercept a and b with the x-axis and y-axis respectively. Let line intersect x-axis at A and y-axis at B.

Coordinates of A will be (a, 0) and B will be (0, b).

Given, sum of intercepts = 7.

∴ a + b = 7 or b = 7 - a.

Equation of AB can be given by two point formula i.e.,

$\Rightarrow y - y$1 = \dfrac{y2 - y1}{x2 - x1}(x - x1) \\[1em] \Rightarrow y - 0 = \dfrac{b - 0}{0 - a}(x - a) \\[1em] \Rightarrow y = \dfrac{b}{-a}(x - a) \\[1em] \Rightarrow -ay = bx - ba \\[1em] \Rightarrow bx + ay - ab = 0 \space …..(i)

Since, line passes through (-3, 8) it will satisfy the above equation. Also, putting b = 7 - a.

$\Rightarrow (7 - a)(-3) + 8a - a(7 - a) = 0 \\[1em] \Rightarrow -21 + 3a + 8a - 7a + a^2 = 0 \\[1em] \Rightarrow a^2 + 4a - 21 = 0 \\[1em] \Rightarrow a^2 + 7a - 3a - 21 = 0 \\[1em] \Rightarrow a(a + 7) - 3(a + 7) = 0 \\[1em] \Rightarrow (a - 3)(a + 7) = 0 \\[1em] \Rightarrow a = 3 \text{ or } a = -7.$

Since, only positive intercepts are made hence a ≠ -7.

b = 7 - a = 7 - 3 = 4.

Putting value of b and a in (i) we get,

⇒ 4x + 3y - 12 = 0.

Hence, the equation of the required line is 4x + 3y = 12.