There are several forms that the equation of a line can take. They may look different, but they all describe the same line--a line can be described by many equations. All (linear) equations describing a particular line, however, are equivalent.
The first of the forms for a linear equation is slope-intercept form. Equations in slope-intercept form look like this:
|y = mx + b|
where m is the slope of the line and b is the y-intercept of the line, or the y-coordinate of the point at which the line crosses the y-axis.
To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope. This is the value of m in the equation. Next, find the coordinates of the y-intercept--this should be of the form (0, b). The y- coordinate is the value of b in the equation.
Finally, write the equation, substituting numerical values in for m and b. Check your equation by picking a point on the line (not the y-intercept) and plugging it in to see if it satisfies the equation.
Example 1: Write an equation of the following line in slope-intercept form:
First, pick two points on the line--for example, (2, 1) and (4, 0). Use these points to calculate the slope: m = = = - .
Next, find the y-intercept: (0, 2). Thus, b = 2.
Therefore, the equation for this line is y = - x + 2.
Check using the point (4, 0): 0 = - (4) + 2 ? Yes.
Example 2: Write an equation of the line with slope m = which crosses the y-axis at (0, - ).
y = x -
Example 3: Write an equation of the line with y-intercept 3 that is parallel to the line y = 7x - 9.
Since y = 7x - 9 is in slope-intercept form, its slope is 7.
Since parallel lines have the same slope, the slope of the new line will also be 7. m = 7. b = 3.
Thus, the equation of the line is y = 7x + 3.
Example 4: Write an equation of the line with y-intercept 4 that is perpendicular to the line 3y - x = 9.
The slope of 3y - x = 9 is .
Since the slopes of perpendicular lines are opposite reciprocals, m = - 3. b = 4.
Thus, the equation of the line is y = - 3x + 4.