A standard line equation is in the form of y = mx + b, where "m" is the slope of the line and "b" is the y-intercept. For example, the graph of y = 2x - 5 has a slope of 2 and a y-intercept of -5. To graph a line, first plot its y-intercept, e.g., (0, -5), and then draw a line through that point with the appropriate slope, e.g., 2.
A system of equations is merely multiple (usually two) line equations. Systems of equations are graphed by graphing each equation separately, as above, and then finding the point of intersection of the two lines. This point of intersection indicates the only pair of (x, y) values at which the equations are both true. For example, the lines y = 7 - 2x and y = 4 - x intersect at (3, 1), and so x = 3 and y = 1.
Inequalities can be graphed as lines. For example, consider the inequality 5 < 2x + y < 7. To isolate "y" as the middle term in the first inequality, subtract 2x from each term. The expression thereby becomes 5 - 2x < y < 7 - 2x, which is really a set of two inequalities: 5 - 2x < y and y < 7 - 2x. Each can be graphed manually or on a graphing calculator. In the example above, the lines would be y = 5 - 2x and y = 7 - 2x. The region between the two lines is the set of solutions to 5 - 2x < y < 7 - 2x.
Systems of inequalities are graphed by graphing each inequality separately as above and then finding the region of intersection of the two solution sets. This overlapping region is the set of all of the solutions to the system of inequalities. For example, the solution sets for the inequalities 5 < 2x + y < 7 and 4 < y - x < 6 overlap in a roughly diamond-like shape. Because these are "more than/less than" (<) inequalities, the lines themselves are not part of the solutions. If the inequalities were instead "more than or equal to/less than or equal to" (symbolized by an underlined version of "<"), the lines would be included in the solutions.