Solving Linear Systems by Graphing

Review: Linear Systems Linear systems are sets of linear equations that describe relationships between two or more variables. Most linear systems involve two or three variables. For the purposes of this packet, we’ll deal primarily with linear systems of two variables.

Graphing Linear Systems Each equation in a linear system of two variables defines a line on the x-y plane. Each of these lines can be graphed individually. The intersection of these lines provides the solution to the system of equations, since that’s the point that satisfies both equations. If they are parallel, there is no solution to the system of equations. If they are the same line, there are infinitely many solutions.

Caution When solving systems of equations by graphing, you must be cautious for a few reasons. First and foremost, it’s for an intersection to look like it occurs at a point with integer coordinates, but actually be slightly off. Second, it’s easy for two lines to look parallel but not quite be parallel. Finally, it can be hard to tell where intersections that do not occur at integer coordinates occur. For this reason, solutions that are obtained through graphing should be treated as estimates.

Example Solve the system of linear equations y = 2x+7 y = 1-x by graphing.

Solution Once we graph the two lines, we see that they intersect at the point (-2, 3). This tells us that the solution to our system is (x, y) = (-2, 3).

Try on your own Solve the following systems of equations by graphing: 1.2x – 3y = –2 4x + y = x - 6y = 12 2x + 2y = 6 3.y = x + 7 y = 15 -3x

Answers 1.(x, y) = (5, 4) 2.(x, y) = (3, 0) 3.(x, y) = (2, 9)

Extension: The Third Dimension It’s much harder to solve systems of equations with three variables by graphing, but still possible. In this case, you’ll have three planes in 3d. The intersection of these three planes is the solution to the system of equations. Let’s try it with the system of equations z = 2x+3y z = 3x - 4y z =-4x-y

Example It’s hard to tell from this picture, but the three planes that represent our three functions intersect at (0,0,0), which tells us that the solution to the system of equations is (x, y, z) = (0,0,0). Generally, it’s easier to just solve the system of equations than graph 3 planes accurately, but the option is still there.