Graphing Linear Equations

Tuesday 4/23/01

Big Idea

Given the equation of a line, you can graph it either by plotting points generated by the equation or by using the slope and intercept found in the equation.

Background / Assumptions

A linear equation is the equation for a line.
The equation for a line has only constants, and either x and/or y multiplied only by constants.
The equation for the line is given to us; we will learn how to derive that later.

Main Explanation

We will look at two ways to graph a line: by points and by "slope intercept".

Graphing a Line by Points

1. Solve the equation for y.

2. Find 3 solution points by plugging in values for x.

Only 2 are needed, but 3 allows for accuracy in checking results.
Use simple x values, like -1, 0, and 1.

3. Plot the 3 points on a graph and draw a line connecting them.

Example: graph y = 2x + 1.
1. The equation is already solved for y: y = 2x + 1.
2. x y
  
  -1 y = 2(-1) + 1 = -2 + 1 = -1
  
  0 y = 2(0) + 1 = 0 + 1 = 1
  
  1 y = 2(1) + 1 = 2 + 1 = 3
3. Plot the points (-1, -1), (0, 1), and (1, 3), and then connect them with a line.
Example: graph 2x + 3y = -12.
1. Solve for y:
  2x + 3y = -12
  3y = -2x - 12
  y = -(2/3)x - 4
2. Choose -3, 0, and 3 for values of x to make things simple.
  
  x y
  
  -3 y = -(2/3)(-3) - 4 = 2 - 4 = -2
  
  0 y = -(2/3)(0) - 4 = 0 - 4 = -4
  
  3 y = -(2/3)(3) - 4 = -2 - 4 = -6
3. Plot the points (-3, -2), (0, -4), and (3, -6), and then connect them with a line.

Graphing a Line Using Slope Intercept Form
1. Solve the equation for y.
2. Determine slope and y intercept. Compare to y = mx + b. m is the slope (rise over run). b is the y intercept (where the line crosses the y axis).
3. Plot b on the y axis (not the x axis).
4. Starting from b, use rise over run (slope) to find another point on the line.
5. Draw a line connecting the two points.

Example: graph 3y - 2x + 4 = 0.
1. Solve for y:
  3y - 2x + 4 = 0
  3y - 2x = -4
  3y = 2x - 4
  y = (2/3)x - 4
2. Compare the equation
  
  y = (2/3)x - 4
  
  y = mx + b
  
  The slope is m = 2/3.
  The y intercept is b = -4.
3. Plot the y intercept (0, -4). (See graph below).
4. Starting at the y intercept, go up 2 units and over 3 units. This leads to the point (3, -2). Plot that point.
5. Draw a line connecting them:

Additional Information

Special case: horizontal lines have the form y = b.
Special case: vertical lines have the form x = a.

Related Topics

How to find the equation of a line...

Tools

Links for further information