Linear Equations

Slope-Intercept-Form. Given the slope m and the y - intercept b of a line ,
an equation for the line can be written as

y = mx + b.

Example. Let a line have slope m = -2 and y-intercept b = 3. Then
the line can be written as y = -2x + 3.

Point-Slope-Form. Given the slope m and a point   on the line, an
equation for the line can be written as



Two-Point-Form. If you know two points   and   on a line,
its equation can be written as



Note that the expression is nothing else but the slope of the
line.

Example. Suppose a line passes through the points
and   Then we can write an equation of the line as



or simplified as :



2 Solving Linear Equations

• Strategy: Move all “x” items to one side of the equation, the rest to
the other side of the equation.

• You are al lowed to do the following:

1. Add ( or subtract ) the same number on both sides.
2. Multiply (or divide) by the same number on both sides. Do not
multiply or divide both sides by 0!

Example 1.

2x - 7 = 4

Add 7 on both sides.



Divide both sides by 2.



Check that is a solution of the original equation!

Example 2.



Add on both sides.



Subtract on both sides.

Simplify both sides.



Multiply both sides by and simplify.



Check thatis a solution of the original equation!

3 Linear Regression : Finding the Best Line
Fitting the Data – An Example

1. Start with the data table, in the example given here, the number of
data points is n = 3:

2. Compute the average ave(x) of the x- coordinates , and the average
ave(y) of the y-coordinates:



3. Compute the deviations from the averages:

4. Compute S_xx and S_xy:

5. The slope m of the regression line is



The y-intercept of the regression line is given by



Thus the regression line has the equation