0.1.3 - Basic Linear Equations

You may recall from an algebra class that the formula for a straight line is \(y=mx+b\) where \(m\) is the slope and \(b\) is the y-intercept. The x-axis is the horizontal axis and the y-axis is the vertical axis.

Axes

The y-intercept is where the line crosses the y-axis.  The slope is a measure of change in y over change in x, sometimes written as \(\frac{\Delta y}{\Delta x}\) where \(\Delta\) ("delta") means change.

Example Section

\(y=2x+1\)

Here, the y-intercept is 1 and the slope is 2.  This line will cross the y-axis at the point (0, 1). From there, for every one increase in \(x\), \(y\) will increase by 2. In other words, for every one unit we move to the right, we will move up 2. 

Slope

In statistics, the y-intercept is typically written before the slope. The equation for the line above would be written as \(y=1+2x\).  

Different notation may also be used:

\(\widehat y = a + bx\), where \(\widehat y\) is the predicted value of y, \(a\) is the y-intercept and \(b\) is the slope.

\(\widehat y = b_0 + b_1 x\), where \(\widehat y\) is the predicted value of y, \(b_0\) is the y-intercept, and \(b_1\) is the slope.