Find the equation of a straight line passing through the point (7, 8) making equal intercepts on both axis.

The equation of a straight line that makes equal intercepts on both the x-axis and y-axis can be derived using the intercept form of a line. The intercept form of a straight line is given by:

$$\frac{x}{a}+\frac{y}{b}=1$$

where $a$ is the x-intercept and $b$ is the y-intercept.

Given that the line makes equal intercepts on both axes, we have: $$a=b$$

Thus, the equation becomes: $$\frac{x}{a}+\frac{y}{a}=1$$

Simplifying, we get: $$x+y=a$$

Now, the line passes through the point $(7,8)$. Substituting this point into the equation, we get: $$7+8=a$$ $$a=15$$

Thus, the equation of the line is: $$x+y=15$$

However, we must also consider the case where the intercepts are equal in magnitude but opposite in sign (i.e., $a=-b$). In this scenario, the equation becomes: $$\frac{x}{a}+\frac{y}{-a}=1$$ $$x-y=a$$

Again, substituting the point $(7,8)$: $$7-8=a$$ $$a=-1$$

Thus, the second possible equation of the line is: $$x-y=-1$$

Final Answer:

The equations of the straight lines passing through the point $(7,8)$ and making equal intercepts on both axes are: $x+y=15$ and $x-y=-1$.