A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is:

Let the line be $\frac{x}{a}+\frac{y}{b}=1$.The intercepts with the axes are $(a,0)$ and $(0,b)$.The midpoint of these intercepts is $\left(\frac{a}{2},\frac{b}{2}\right)$.Given that the point $A(3,4)$ bisects the intercept:$\frac{a}{2}=3⟹a=6$$\frac{b}{2}=4⟹b=8$ The equation of the line is $\frac{x}{6}+\frac{y}{8}=1$.Multiplying by $24$:$4x+3y=24$.