Sections 2.4 & 2.5 - Lines and Linear Equations in Two Variables

Section Objectives

  1. Determine solutions of two-variable linear equations.
  2. Graph a line by finding two points on the line.
  3. Find the - and -intercepts of a line.
  4. Compute the slope of a line and interpret it as a rate of change.
  5. Identify equations of horizontal or vertical lines and graph them.



Equations of Lines

Any linear equation in two variables can be written in the form

where and are not both zero. The form is called standard form.



A solution of a two-variable linear equation is a point that satisfies the equation. Every two-variable linear has infinitely many solutions. If all such solutions are plotted in a rectangular coordinate system, the resulting graph is a line.




Examples










Graphing Lines

Remember from geometry that a line is uniquely determined by two distinct points. Therefore, one approach (among others that we will learn) to graphing the line described by a two-variable linear equation is to:

  1. determine two solutions (i.e, two points on the line),
  2. plot them, and
  3. sketch a line through them.

Some of us may also like to find one more solution to check our work.



Examples













Intercepts

Points of the form and are called - and -intercepts, respectively. These are points where a graph intersects the -axis or the -axis.



In problem situations, Intercepts often have some kind of physical significance. They are usually worth finding! And they can be useful for graphing.



Examples







Slope

The slant of a line is measured by a number called slope. Before we define slope, here are three important ideas to keep in mind:

  1. Slope is only defined for non-vertical lines.
  2. The slope of a line is the same at any point on the line.
  3. The slope is a measure of the rate of change of with respect to .


Suppose and are any two distinct points on a non-vertical line. The slope of the line is given by .




Examples







Parallel and Perpendicular Lines

Two non-vertical lines are parallel if and only if they have the same slope.


Two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals (that is, the product of their slopes is ).


What if one or both of the lines are vertical?




Examples







Miscellaneous Examples