Slope-Intercept Form

• The slope-intercept form of a linear equation is y = mx + b, where "m" is the slope and "b" is the y-intercept. The y-intercept is the point at which the graphed line intersects with the y-axis. The slope provides a way to find a point on the line based on the point prior. A positive slope is applied using "rise over run," meaning a number of spots moved to the right, then a number of spots moved up.
  
  

Point-Slope Form

• Using the slope-intercept form requires knowing the y-intercept. If the y-intercept is not the point that is known, the point-slope form can be used. The point-slope form states y - k = m(x - h) where "m" is still the slope and "h" and "k" represent the nonintercept point of (h, k). Solving this form for "y" puts it into slope-intercept form.
  
  For example, for a line with a slope of 3 and a point of (2, 4): y - 4 = 3(x - 2). Distribute the 3: y - 4 = 3x - 6. Add 4 to both sides: y = 3x - 2.
  
  

Two-Point Form

• If the slope and y-intercept are both unknown, but two other points are available, the slope-intercept form can still be reached. The formula for a slope, on its own, is m = (y1 - y2) / (x1 - x2) for points (x1, y1) and (x2, y2). The two-point form takes this a step further, stating y - y1 = ((y1 - y2) / (x1 - x2)) * (x - x1).
  
  Use an example line with the points (2 , 3) and (4, 5): y - 3 = ((3 - 5) / (2 - 4)) * (x - 2). Simplify: y - 3 = ((-2)/(-2)) * (x - 2) or y - 3 = 1 * (x - 2). Distribute the 1 through the parenthesis: y - 3 = x - 2. Add 3 to both sides to put in slope intercept form, with a slope of 1: y = x + 1.
  
  

Graphing

• Linear equations are solved through graphing. Graphing by hand requires finding several points to place and form the basis for the line. If the information given includes points, those may be included in the graphed points. But the easiest way to find additional points is using the y-intercept and the slope.
  
  For example, in the linear equation y = 5x + 6, the y-intercept is 6, or point (0, 6). The slope is 5, or 5/1, which can be applied to the y-intercept by moving five spots right on the x-axis and one spot up on the y-axis: (0 + 5, 6 + 1) = (5 , 7). Apply the slope to the new point to find another point: (5 + 5, 7 + 1) = (10, 8). Repeat the process: (10 + 5, 8 + 1) = (15, 9). Repeat for the fifth point: (15 + 5, 9 + 1) = (20, 10).