Find the equation of a line pasing through the point (2,2) and sum of the intercepts is 9?

Answer 1

The equation of the line is y = ax + b

At the point 2,2: 2 = 2a + b, so b = 2 - 2a

The y intercept occurs at x = 0, so int(y) = b
The x intercept occurs y = 0, so 0 = ax + b and int(x) = -b/a

The sum int(y) + int(x) = b - b/a = 9.

Plug in b = 2 - 2a to get 2 - 2a - 2/a + 2 = 9

2*(a + 1/a) = -5

2a^2 + 5a + 2 = 0

solve for a, then put into b = 2 - 2a to get b

a = (-5 ± √[25 - 4*2*2]) / 4 = (-5 ± 3)/4

a = -2 or -1/2

SInce b = 2 - 2a,

for a = -2, b = 2 + 4 = 6
for a = -1/2, b = 2 + 1 = 3

The equations are
y = -2x + 6 or y = -(1/2)x + 3

Check: set x = 2: this gives y = 2 for both eqs.

y intercept is 6 or 3
x intercept is 3 or 6

in either case the sum is 9. So both equations work.