Step 1

Equation of the line in the intercept form is

$ \large\frac{x}{a}$$+\large\frac{y}{b}$$=1$-----(1)

It is given that the sum of the intercepts is 14.

(i.e) $a+b=14$

or $ b = 14-a$

Substituting this in equivalent (1) we get.

$\large\frac{x}{a}$$+\large\frac{y}{14-a}$$=1$

(i.e) $x(14-a)+y(a)=a(14-a)$

Step 2

This line passes through the point (3, 4)

$ \therefore 3(14-a)+4(a)=a(14-a)$

$ \Rightarrow 42-3a+4a=14a-a^2$

(i.e) $a^2-13a+42=0$

On factorizing this we get,

$(a-6) (a-7)=0$

$ \therefore a = 6 $ or $ a = 7$

Step 3

Case (i)

If $ a = 6$ then $ b = 8$

Hence the equation of the line is

$ \large\frac{x}{6}$$+\large\frac{y}{8}$$=1$

(i.e) $8x+6y=48$

or $4x+3y=24$

Step 4

Case (ii)

If $ a=7$ then $b = 7$

Hence the equation of the line is

$ \large\frac{x}{7}$$+\large\frac{y}{7}$$=1$

$ \Rightarrow x+y=7$