Equation of a line: The derivation of y = mx + b

We have discussed in context the origin (click here and here) of the linear equation y = ax + b, where a and b are real numbers.  We have also talked about the slope of a line and many of its properties. In this post, we will discuss the generalization of the equation of a line in the coordinate plane based on its slope and y-intercept.

We have learned that to get a slope of a line, we only need two points.  We have also learned that given two points on a line, its slope is described as the rise (difference in the y-coordinates) over the run (difference in x-coordinates).  Therefore, if we have two points with coordinates (x_1,y_1) and (x_2,y_2), the slope m is  defined the formula

m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}.

All the points on a vertical line have similar x-coordinates; therefore, the run {x_2 - x_1} is equal to 0 making m undefined.  From here, we can conclude a vertical line has no slope. » Read more

Matchsticks, Linear Relations, and Multiple Representations


We have mentioned the different types of functions in the Introductions to Functions post.  In this post, we are going to learn about linear function and its  characteristics.

To start, let us examine the problem below taken from the TIMSS 2003 released items given to Grade 10 students in more than 40 countries all over the world.

Matchsticks are arranged as shown in the figures.

If the pattern is continued, how many matchsticks would be used to make figure 10.

A. 30                      B. 33                      C. 36                      D. 39                      E. 42

The problem is too easy that even a first grade pupil would be able to answer it given enough time. Smart students would be able to easily see patterns. For example, they can relate the number of squares to the number of matchsticks.  If they cannot find a pattern, the last resort would be by brute force; that is, by manually drawing the tenth figure. » Read more