- The coordinate plane.
- Graphing lines on the coordinate plane.
- Slope and y-intercept for linear equations.
- Slope-intercept form for linear equations,
**y**= m**x**+ b. - Point-Slope form for linear equations,
**y**– y_{1}= m (**x**– x_{1}). - Equations of parallel lines have the same slope.
- Equations of perpendicular lines have either
- one being horizontal and the other vertical, or
- slopes that multiply to give -1

The

The first example will be the most complex. We start with

Another example would be to start with

A third example is the slope-intercept form

We have seen that we can transform slope-intercept form equations into standard form equations. But why should we want to do this? There are a number of reasons. First, standard form allows us to write the equations for vertical lines, which is not possible in slope-intercept form. Remember that vertical lines have an undefined slope (which is why we can not write them in slope-intercept form). However, the vertical line through the point (4,7) has the standard form equation

A second reason for putting equations into standard form is that it allows us to employ a technique for solving systems of linear equations. This topic will not be covered until later in the course so we do not need standard form at this point. However it will become quite useful later.

A third reason to use standard form is that it
simplifies finding parallel and perpendicular lines.
Let us look at the typical parallel line problem.
*Find the equation of the line that is parallel to the line 3 x+4y=17
and that contains the point (2,8).*
The usual approach to this problem is to find the slope of the given line and then to
use that slope along with the given point in the point-slope form for a linear
equation.
However, if we look at the standard form of a linear equation,

We have seen that parallel lines in the standard form

*"Find the equation of the line that is perpendicular to the line
2 x+^{– }5y=^{– }19
and that contains the point (4,^{– }7)."*
The answer must look line

From the presentation above we can see that we can do the "usual" parallel and perpendicular line problems in just a few steps without ever finding the slope of the original line and without using the point-slope form of a linear equation.

For completeness we should check our methods with horizontal and vertical lines.
The standard form for a horizontal line is 0**x**+1**y**=C. Another
horizontal line, one parallel to the first, will still have the form
0**x**+1**y**=D. Therefore, our rule for finding parallel lines will still
work. We just leave the "A" and "B" values the same and find a new value for "D"
by substituting the coordinates of the external point.

The standard form for a vertical line is 1**x**+0**y**=C. Another
vertical line, one parallel to the first, will still have the form
1**x**+0**y**=D. Therefore, our rule for finding parallel lines will still
work. We just leave the "A" and "B" values the same and find a new value for "D"
by substituting the coordinates of the external point.

If we start with a horizontal line in the form 0**x**+1**y**=C, and we reverse
the "A" and "B" values, and reverse the sign of one of them, we get
1**x**+0**y**=D, which is the general form of a vertical line. This
corresponds to our method for finding perpendicular lines.

**PRECALCULUS: College Algebra and Trigonometry**

© 2000 Dennis Bila, James Egan, Roger Palay