SLE - Solve Systems of Equations Graphically (Lesson)

Systems of Equations

What exactly is a system of linear equations? Well, a system of equations is two or more linear equations that show a relationship among the given variables. For now, we'll only focus on systems that have two equations. Just know that the system can have more than two equations.

Here is a system of linear equations:

2x - y = -9

x + y = 6

We can solve a system by several methods. The methods we'll discuss are as follows:

  • Graphing Method
  • Substitution Method
  • Elimination Method

What exactly are we finding when we solve a system of equations? Well, when we solve a system, we are looking for that one point, or ordered pair, that makes both equations true. Graphically, that means that we are looking for the point where the lines of each equation intersect. If we are given the solution, then, we need to plug the numbers into each equation and check to make sure the solution is correct.  We should do this for all methods.

Special Cases

In most instances, we will run across situations where there is only one solution....that one point where the graphs intersect. But, just like when we are solving equations, there are those situations where we get a "no solutions" or an "infinitely many solutions" answer. How would that happen with a system? Well, let's think. What is the only time you can have two lines that would never touch or intersect?

If you said when they are parallel, that's exactly right!!

If you have parallel lines, then there is no solution to the system because those two lines will never, ever intersect, or share a point. This means no ordered pair will ever satisfy both equations.

Now, let's investigate the other special situation of having infinitely many solutions. If the system has infinitely many solutions, then they share infinitely many ordered pairs. How might that happen? Well, it would happen if the lines were co-linear. In other words, if they are the same line, they share each of their points.

Solve Systems of Equations Using the Graphing Method

Think of a system with two equations like those below.

 If you graph each of the two equations onto the same coordinate graph,  then the point where the lines intersect is the solution to this system.

Below are two different ways to graph linear equations:

  • Slope Intercept - When your equation is in the form y = mx + b, this is the best method to use.
    • y = 2x + 1 and y = 6x - 11
    • For y = 2x + 1, start at the y-intercept (0, 1) and move up and over according to the slope (up 1, right 1). Connect the dots and extend your line.
    • For y = 6x - 11, start at the y - intercept (0, -11) and move up 6 and to the right 1. Connect the dots and extend your line. 
    • The solution will be the point where both lines intersect.
  • Standard Form - This is a good graphing method to use if your equation is in the form: Ax + By = C.
    • 3x + y = 6 and 2x + y = 14
    • Create an intercept chart to plot two points and connect and extend the line.  You only need two points to draw a line. 
    • To create an intercept chart, plug in 0 for x to find the y - intercept and plug in 0 for y to find the x - intercept. That will give you two points: (0, 14) and (7, 0)
      • 3(0) + y = 6; 3x + 0 = 6 and 2(0) + y = 14; 2x + 0 = 14

When should I use graphing?

Graphing can be good to use if you can easily graph your equations and if your lines intersect at specific whole number coordinates. However, without a graphing calculator or computer, this method is often the least accurate. It is awesome for estimates though!

Example

What does it look like?

  • System of Equations: y = 2x - 5 and 3x - y = 7

Here is the graph of these equations. We can see that the lines intersect at (2, -1). This is the solution, and the one ordered pair that satisfies both equations. We can verify this by plugging the coordinates into each equation to check:

Graph of two lines that intersect a two negative one

Equation One

  • y = 2x - 5
  • -1 = 2(2) - 5
  • -1 = 4 - 5
  • -1 = -1

Equation Two

  • 3x - y = 7
  • 3(2) - (-1) = 7
  • 6 + 1 = 7
  • 7 = 7

Watch this video to see a few more examples!

 

Note that there are three different options for how lines can intersect, which means that there are three different types of answers you might give.

Graphing Systems of Equations

 

Graphing Systems of Equations Practice

Try these problems to see if you've got it.

1. Is (1, -3) a solution to the system of equations: y = 4x + 3 and y = -x - 2 

2. Is (2, 2) a solution to the system of equations: y = 3x - 4 and y = -(1/2)x + 3 

3. On your own paper, graph the system of equations and identify the solution: y = -x + 1 and y = (1/3) - 3 

4. On your own paper, graph the system of equations and identify the solution: 6x + y = -3 and x + y = 2 

5. On your own paper, graph the system of equations and identify the solution: 2x + 3y = -9 and y = -x - 2 

6. On your own paper, graph the system of equations and identify the solution:

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Practice

If you would like to practice more problems and check your work, click here. Links to an external site. Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

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