Chapter 4.2, Problem 52HP

To determine

What information is required to form an equation line.

Answer to Problem 52HP

We need either one of the following to form equation of line:

1. Slope and y-intercept of line.
2. Any two points that lie on the line.
3. Slope and any point that lies on the line .

Explanation of Solution

Formula Used:

The slope intercept form of a line is :

  $\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\\ if\\ (x_1,y_1),(x_2,y_2)=po\mathrm{int}s\_on\_line\\ m=\frac{y_2-y}{x_2-x_1}\end{array}$

Calculation:

1. If we are given slope and y-intercept, we can directly form the slope-intercept form of line.
2. If we are given two points on line , we can find the slope and y-intercept of line as follows:

  $\begin{array}{l}y=mx+b\mathrm{...........................}[slope-\mathrm{int}ercept]\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\\ if\\ (x_1,y_1),(x_2,y_2)=po\mathrm{int}s\_on\_line\\ m=\frac{y_2-y}{x_2-x_1}\\ for\_y-\mathrm{int}ercept:\\ y=mx+b\\ \Rightarrow y_1=(\frac{y_2-y}{x_2-x_1})x_1+b\mathrm{.............}[(x_1,y_1)lie\_on\_line]\\ \Rightarrow b=y_1-(\frac{y_2-y}{x_2-x_1})x_1\end{array}$

Hence, we can then substitute the values of slope and y-intercept in the general slope-intercept form $y=mx+b$ to get the required equation of line.

3. If we are given slope of line and one point that lies on line , we can easily find the y-intercept as in part 2. and substitute the values in general equation of slope-intercept form $y=mx+b$ to get the required equation of line.