5.02 Graph systems of equations

If we have two equations with the same two variables in them (x \text{ and } y), then we call them a system of equations. They are also commonly referred to as simultaneous equations.

We might want to find a common pair of x and y values that satisfy both of these equations simultaneously. If we can find any values of x and y that successfully do this, then we will have found a unique solution to our system.

One way to find a solution common to each equation is to graph them and identify the point of intersection (where the two graphs cross). This point will be a solution to both equations and therefore solve our system of equations.

With this in mind, how many solutions should be possible for a system of linear equations?

• One solution: If the lines representing the two equations are not parallel, then there should be exactly one point of intersection between them (as pictured below). Why do you think this would be true?

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  An example is the system of equations:

  y = \dfrac{x}{2} - 2 \\ y = -x + 7 which has a graph of two intersecting lines and the point where the two lines intersect is the only solution.

• No solution: If the lines are parallel and distinct, then there will not be any points of intersection between them. This means no corresponding x and y values satisfy both equations simultaneously.

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  Example is the graph of the system of equations:

  y = \dfrac{x}{2} - 2

  y = \dfrac{x}{2} + 7 in which the lines are parallel.

• Infinitely many solutions: The final case is when two different equations have the same graphical representation. For example, if the graphs of x + y = 5 and 2x + 2y = 10 were placed on the same set of axes, we would end up with two lines lying perfectly on top of one another. So every point on the line is a point of intersection, meaning there are infinitely many solutions to this system of equations.

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  Another example is the graph of the system of equations y = \dfrac{x}{2} - 2 \text{ and } y = \dfrac{x}{2} + 7 whose the lines are coinciding.

Let's also remember that to find the slope of a linear equation all we have to do is put it in slope-intercept form y = mx + b and m will be our slope. This means that for example, the system of linear equations y = 3x - 1 \text{ and } y = 3x - 6 will never have a solution since they both have a slope of 3.

Examples