7-1 Graphing Systems of Equations


Two equations together are called a system of equations. A solution of a system of equations is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have 0, 1 or an infinite number of solutions.

  • If the graphs intersect or coincide, the system of equations is said to be consistent. That is, it has at least one ordered pair that satisfies both equations.
  • If the graphs are parallel, the system of equations is said to be inconsistent. There are no ordered pairs that satisfy both equations.
  • Consistent equations can be independent or dependent. If a system has exactly one solution, it is independent. If the system has an infinite number of solutions, it is dependent.

systems.jpg

Example 1: Use the graph below to determine whether each system has no solution, one solution, or infinitely many solutions.
graphin_soltns.jpg


  1. y = 2x + 3 and y = 2x - 4
  2. 6x - 3y = 12 and y = 2x - 4
  3. 6x - 3y = 12 and y = -x + 1

Answers:
  1. This system has no solution because the lines are parallel, which means they will never intersect. I know that the lines are parallel because they have the same slope.
  2. This system has infinitely many solutions because the lines are the same line. I know that the lines are the same because I can put them both into slope intercept form and see that they are identical.
  3. This system has one solution because the lines intersect. Any two lines that have different slopes are bound to intersect somewhere on the coorinate plane. Therefore, if I am ever given a system of equations where each line has a different slope, I know there will be one solution.

Solve Systems by Graphing

One method of solving systems of equations is to carefully graph the equations on the same coordinate plane.

Example 2a: Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

  • y = -x + 8
  • y = 4x - 7
35_soltn.jpg

Remember, to graph a line, we must write the equation in slope-intercept form. We then start at the y-intercept and count our slope by using rise over run.

After graphing my system, I found the solution to be the point (3, 5). This is the solution because the point (3, 5) is the only point that lies on both lines. Therefore, if I were to plug in 3 for x and 5 for y into either equation, I will obtain true statement. This is a great method to use to check a solution.

The point of intersection of a system of equations is always the solution of the system.

Example 2b: Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

  • x + 2y = 5
  • 2x + 4y = 2

In order to solve a system by graphing, I must make sure both of my equations are in slope-intercept form. In this example, I need to solve both equations for y before I can graph.

eq1a.jpg eq2.jpg
Now, I can graph my system since both equations are in slope-intercept form.

nosoltnex2b.jpg


The lines in this system are parallel. They will never intersect so there is no solution. There is not one ordered pair that will satisfy both equations.

More Practice:





Helpful Links:
Solve by Graphing - Purple Math
Solve by Graphing - Regents Prep