Answer
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Work Step by Step
Replacing $\left( {{x}_{1}},{{y}_{1}} \right)=\left( a,0 \right)\ \text{ and }\ \left( {{x}_{2}},{{y}_{2}} \right)=\left( 0,b \right)$ in the slope-intercept equation, the equation of the line is obtained as below:
$\begin{align}
& y-0=\frac{b-0}{0-a}\left( x-a \right) \\
& y=\frac{-b}{a}\left( x-a \right) \\
& y=\frac{-b}{a}x+b \\
& \frac{y}{b}=\frac{-1}{a}x+1
\end{align}$
Rearranging the above equation, we get,
$\frac{x}{a}+\frac{y}{b}=1$
Hence, the equation of the line passing through the points $\left( a,0 \right)$ and $\left( 0,b \right)$ can be written in the form $\frac{x}{a}+\frac{y}{b}=1$
Now, we can write $\frac{x}{a}+\frac{y}{b}=1$ as
$\begin{align}
& \frac{x}{a}+\frac{y}{b}=1 \\
& \frac{y}{b}=-\frac{x}{a}+1 \\
& y=\frac{-b}{a}x+\frac{1}{b} \\
\end{align}$
Hence, $\frac{x}{a}+\frac{y}{b}=1$ is called the intercept form of the line as $a$ is the x-intercept and $b$ is the y-intercept.
