Introduction to Linear Equations

Slope Intercept Form

• Graphing a linear equation requires that it is placed in slope intercept form. Slope intercept form states that y = mx + b, where "y" and "x" are the variables, "m" is the slope of the line and "b" is the y-intercept, or the point at which the line crosses over the y-axis. Placing an equation in this form requires that the slope and y-intercept are provided in the problem.

Y-Intercept

• The y-intercept is the point at which the line intersects the y-axis, which is the vertical axis on the graph. The intercept can be represented as a graphical point, where the x value is always 0 and the y value is the given "b" value. For example, the equation y = 3x + 4 would have a y-intercept of 4 or point (0, 4).

Point Slope Form

• If the y-intercept isn't known, the equation can't be put into slope intercept form. But if the slope and one point on the graph, (x1, y1), are known, then you can use the point slope form to put the equation into slope intercept form. The point slope form states y - y1 = m(x - x1).

  For example, for a line with a slope of 3 and a point of (2, 5): y - 5 = 3(x - 2). Distribute the 3: y - 5 = 3x - 6. Add 5 to both sides: y = 3x - 1. The slope is 3 and the y-intercept is -1 or (0, -1).

Slope

• The slope of a line is the difference between one point, (x1, y1), and the next point on the line, (x2, y2). The difference is represented as (y2 - y1) / (x2 - x1). The slope is often described as being "rise over run," meaning that it represents movement on the y-axis followed by movement on the x-axis.

  For example, in the equation y = 5x + 3 the slope is 5 or 5/1. That means the points will move 5 spaces up the y-axis followed by 1 spot over on the x-axis. Using the y-intercept as an example point, the slope can be applied like so: (0 + 1, 3 + 5) = (1, 8). This is a handy method of finding additional points for the line for graphing.

Two Point Form

• If the slope and y-intercept are unknown, the slope intercept form can still be found if two points, (x1, y1) and (x2, y2), are given. The two point form is simply the point slope form with the definition of a slope substituted in for the "m". The two point form states: y - y1 = ((y2 - y1) / (x2 - x1)) * (x - x1).

  Practice with a line that includes the points (4, 8) and (2 , 7). Fill in the known information: y - 8 = ((7 - 8 / 2 - 4)) * (x - 4). Simplify, starting with the slope: y - 8 = (1/2) * (x - 4). Distribute the (1/2): y - 8 = (1/2)x - 2. Add 8 to both sides: y = (1/2)x + 6. The slope is (1/2) and the y-intercept is 6 or point (0, 6).