Expert: Josh - 4/2/2008

Question
find the equation of the line perpendicular to 2x+3y=4 through the point (6,5)

thanks

Answer
Imran,

Here are the steps:

1) Determine the gradient or slope of the given straight line equation.
2) Calculate the gradient or slope of the perpendicular line.
3) Substitute the point where the perpendicular line passes through to obtain the final equation.

Theorem: If "m" represents the slope of the given line, and "n" represents the slope of the perpendicular line (one which is at 90 degrees to the given line), then m*n = -1.

Solution:
1) We turn 2x+3y=4 into the slope-intercept form (y=mx+b), writing it as y=(-2x+4)/3. We identify the slope as m=-2/3.

2) Using the theorem, m*n=-1, the slope of the perpendicular line must be n=-1/m=3/2. Hence, the perpendicular line has an equation in the form of y=nx+c, where c is the y-intercept (i.e., where the line crosses the y axis).

3) y=(3/2)x+c actually represents a family (many possible) perpendicular lines to the given one. The one we look for must pass through (x=6,y=5). i.e., (6,5) must satisfy the equation y=(3/2)x+c. Substituting x=6,y=5, you work out the value of the constant "c". This gives the final equation. I'll let you complete this yourself.