Chapter 5.4, Problem 23MCQ

To determine

To describe the two methods that can be used to write an equation of a line given its graph.

Explanation of Solution

The two methods are:- Point-slope form of a linear equation:

The point slope form of an equation of a non-vertical line with slope $m$ and through point $(x_1,y_1)$ is $y-y_1=m(x-x_1)$

For example:

Suppose slope $m=2$ and points $(x_1,y_1)=(2,3)$

The equation is in point-slope form,

  $\begin{array}{l}y-y_1=m(x-x_1)\\ y-3=2(x-2)\\ y-3=2x-4\\ y-2x+1=0\end{array}$

Then, sketch the graph for the above equation.

Slope intercepts form of a linear equation:

The slope-intercept form of a linear equation of a non-vertical line is $y=mx+b$

Here m is the slope.

And b is the y-intercept.

For example:

Consider the points on the line $(0,-1)$ and $(2,0)$ .

Sketch the graph as follows:

Find the slope and y-intercept; use the slope intercept form,

  $\begin{array}{l}\text{slope}=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{0+1}{2-0}\\ m=\frac{1}{2}\end{array}$

Substitute the slope $m=\frac{1}{2}$ and y-intercept $b=-1$ into the slope-intercept form.

Use slope intercept form

  $\begin{array}{l}y=mx+b\\ y=\frac{1}{2}x-1\end{array}$

Therefore, the equation of the line passing through the points $(0,-1)$ and $(2,0)$ in slope intercept from is $y=\frac{1}{2}x-1$