Chapter 3.4, Problem 57WE

To determine

To prove:Show thatan equation of the line having $x-\text{intercept }a$ and $y-\text{intercept }b$ is $\frac{x}{a}+\frac{y}{b}=1$ .

Answer to Problem 57WE

Therefore, the intercept form of line $\frac{x}{a}+\frac{y}{b}=1$ .

Explanation of Solution

Given information: The equation of the line is called intercept form,

  $\frac{x}{a}+\frac{y}{b}=1$

Formula used: For calculation of slope passing through two points,

  $\text{Slope,}\left(m\right)=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$

Ecplanation: Let a arbitrary point $(x,y)$ on the line $\frac{x}{a}+\frac{y}{b}=1$ .

  $\text{Here }a\text{ is }x-\text{intercept and }b\text{ is }y-\text{intercept}\text{.}$ The equation can be represented graphically as,

  

To find the slope of line passing through points,

  $\begin{array}{l}\left(a,0\right)\text{ and }\left(x,y\right)\\ \text{Slope,}\left(m\right)=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\\ m=\frac{\left(y-0\right)}{\left(x-a\right)}\end{array}$

  $\begin{array}{l}\left(0,b\right)\text{ and }\left(a,0\right)\\ \text{Slope,}\left(m\right)=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\\ m=\frac{\left(0-b\right)}{\left(a-0\right)}\\ m=-\frac{b}{a}\end{array}$

Slope for both will be the same because it belongs to same line.

  $\begin{array}{l}\frac{y-0}{x-a}=-\frac{b}{a}\\ ay=\left(-b\right)\times \left(x-a\right)\\ ay=-bx+ab\\ bx+ay=ab\\ \text{Dividing both side by }ab\text{ we get,}\\ \frac{x}{a}+\frac{y}{b}=1\end{array}$

Therefore, the intercept form of line $\frac{x}{a}+\frac{y}{b}=1$ .