Answer
Given: The line has x-intercept $x=a$ while y-intercept $y=b$.
We need to show the equation of line is $\dfrac{x}{a}+\dfrac{y}{b}=1$.
Using the given we know that points $(a, 0)$ and $(0, b)$ lie on the line.
Thus, the slope of the line using two points is $m=\dfrac{b-0}{0-a}=\dfrac{-b}{a}$.
Now, the equation of line using slope $m$ and a point on line $(a, 0)$ is given by
$y-0=m(x-a)$
$\Rightarrow y=\dfrac{-b}{a}(x-a)$
$\Rightarrow y=\dfrac{-b}{a}x+b$
$\Rightarrow \dfrac{b}{a}x+y=b$
$\Rightarrow \dfrac{x}{a}+\dfrac{y}{b}=1$ on dividing throughout by $b$.
Hence, the equation of the line is $\dfrac{x}{a}+\dfrac{y}{b}=1$ as required. Hence, proved.
Work Step by Step
The intercepts are the points at which the line intersects axes, so, the value of intercepts gives points on the line and these points can be used to derive the required form of equation as shown above.
