Chapter B, Problem 46E

(a)

To determine

To calculate: To show that the equation of line can be put in the form $\frac{x}{a}+\frac{y}{b}=1$

Answer to Problem 46E

It is proved thatequation of line can be put in the form $\frac{x}{a}+\frac{y}{b}=1$

Explanation of Solution

Given information: $a$ and $b$ are x-intercepts and y-intercepts respectively

Formula Used:

Slope of the line passing through the points $P_1\left(x_1,y_1\right)$ to $P_2\left(x_2,y_2\right)$ is

  $m=\frac{y_2-y_1}{x_2-x_1}$

Equation of line having slope $m$ and passing through the point $P_1\left(x_1,y_1\right)$ is

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Given that $a$ and $b$ are x-intercepts and y-intercepts respectively

Thus, points are as follows:

  $\left(a,0\right)$ and $\left(0,b\right)$

Slope of line passing through above two points is calculated as

  $\begin{array}{l}m=\frac{0-b}{a-0}\\ m=-\frac{b}{a}\end{array}$

Now, equation of line passing through $\left(a,0\right)$ is calculated as

  $\begin{array}{l}y-0=-\frac{b}{a}\left(x-a\right)\\ y=-\frac{b}{a}x+b\\ \frac{y}{b}=-\frac{1}{a}x+1\\ \frac{y}{b}=-\frac{x}{a}+1\\ \frac{x}{a}+\frac{y}{b}=1\end{array}$

Conclusion:

Hence, proved that equation of line can be put in the form $\frac{x}{a}+\frac{y}{b}=1$

(b)

To determine

To calculate: To find the equation of line

Answer to Problem 46E

The equation of line is $\frac{x}{6}-\frac{y}{8}=1$

Explanation of Solution

Given information: x-intercept is $6$ and y-intercept is $-8$

Formula Used:

If $a$ and $b$ are x-intercept and y-intercept respectively, then equation of line can be put in the form $\frac{x}{a}+\frac{y}{b}=1$

Calculation:

Given x-intercept is $6$ and y-intercept is $-8$

Thus,

  $\begin{array}{l}a=6\\ b=-8\end{array}$

Equation of line in two-intercept form is given as

  $\frac{x}{a}+\frac{y}{b}=1$

Substituting the values,

  $\begin{array}{l}\frac{x}{6}+\frac{y}{-8}=1\\ \frac{x}{6}-\frac{y}{8}=1\end{array}$

Conclusion:

Hence, the equation of line is $\frac{x}{6}-\frac{y}{8}=1$