Summarize:

Unless you are using a computer or graphing calculator, many systems of equations can only be approximately solved using this method. Your teacher or math textbook may require you to use this method so you are familiar with graphing equations as lines. You can also use this method to double-check your answers from one of the other methods. The basic idea is to graph both equations, and find the point where they intersect. The x and y values at this point will give us the value of x and the value of y in the system of equations. Keeping the two equations separate, use algebra to turn each equation into the form "y = __x + __". For example:  Your first equation is 2x + y = 5. Change this to y = -2x + 5. Your second equation is -3x + 6y = 0. Change this to 6y = 3x + 0, then simplify to y = ½x + 0.  If both equations are identical, the entire line will be an "intersection". Write infinite solutions. On a piece of graph paper, draw a vertical "y axis" and a horizontal "x axis." Starting at the point where they intersect, label the numbers 1, 2, 3, 4, etc. moving up on the y-axis, and again going right on the x-axis. Label the numbers -1, -2, etc. moving down on the y-axis and left on the x-axis.  If you don't have graph paper, use a ruler to make sure the numbers are spaced precisely apart. If you are using large numbers or decimals, you may need to scale your graph differently. (For example, 10, 20, 30 or 0.1, 0.2, 0.3 instead of 1, 2, 3). Once you have an equation in the form y = __x + __, you can start graphing it by drawing a dot where the line intercepts the y-axis. This is always going to be at a y-value equal to the last number in this equation.  In our examples from earlier, one line (y = -2x + 5) intercepts the y-axis at 5. The other (y = ½x + 0) intercepts at 0. (These are points (0,5) and (0,0) on the graph.) Use different colored pens or pencils if possible for the two lines. In the form y = __x + __, the number in front of the x is the slope of the line. Each time x increases by one, the y-value will increase by the amount of the slope. Use this information to plot the point on the graph for each line when x = 1. (Alternatively, plug in x = 1 for each equation and solve for y.)  In our example, the line y = -2x + 5 has a slope of -2. At x = 1, the line moves down 2 from the point at x = 0. Draw the line segment between (0,5) and (1,3). The line y = ½x + 0 has a slope of ½. At x = 1, the line moves up ½ from the point at x=0. Draw the line segment between (0,0) and (1,½).  If the lines have the same slope, the lines will never intersect, so there is no answer to the system of equations. Write no solution. Stop and look at your graph. If the lines have already crossed, skip ahead to the next step. Otherwise, make a decision based on what the lines are doing:  If the lines are moving toward each other, keep plotting points in that direction. If the lines are moving away from each other, move back and plot points in the other direction, starting at x = -1. If the lines are nowhere near each other, try jumping ahead and plotting more distant points, such as at x = 10. Once the two lines intersect, the x and y values at that point are the answer to your problem. If you're lucky, the answer will be a whole number. For instance, in our examples, the two lines intersect at (2,1) so your answer is x = 2 and y = 1. In some systems of equations, the lines will intersect at a value between two whole numbers, and unless your graph is extremely precise it will be difficult to tell where this is. If this happens, you can write an answer such as "x is between 1 and 2", or use the substitution or elimination method to find the precise answer.
Only use this method when told to do so. Solve both equations for y. Draw coordinate axes. Draw the y-intercept for each line. Use the slope to continue the lines. Continue plotting the lines until they intersect. Find the answer at the intersection.