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So equal intersept mean tan45 = m = 1

do the equation of line 

(y - y₁)=m(x - x₁)    where y₁ = -3 , x₁ = 5 , m = 1

(y+3)=1(x - 5)
x - y = 8 ANSWER
the answer is correct souvik.
you can get x+y=2 by taking tan(theta)=-1 which you have missed.
there are two answers.
Comment has been deleted
you will get by both....
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Let the intercept be m
let the points be (0,m) and (m,0)
slope = (0-m)/(m-0) = -1
the line is
(y-m) = -1*(x-m)

it passes through (5,-3)
5-m = -1*(-3-m)

line is y-1 = -1*(x-1)
=>y-1 = 1-x

1 5 1
i have given the solution for the answer you required.
to get the other solution, take the point (0,-m) and (0,m).