Download Systems of Equations Guided Notes on the Graphing Method

Systems of Equations
Guided Notes on the Graphing Method
Definition: Two or more equations with the same variables from a system of equations.
There are three methods that we will use to solve a system of equations. These three methods are the
graphing method, the substitution method, and the elimination method.
Recall: Remember from the graphing unit that the graph of a line is the set of all solutions of that line. In
other words, any point (x, y) you pick on a line, this point is a solution to the linear equation that is being
graphed. To solve a system of equations, you must find all ordered pairs (x, y) that make both equations
true.
Example: Suppose we have a system of equations made up of the following two equations:
y = -2x + 3
y = 4x – 3
The graph of these two lines is shown below on the same graph.
The solution to the system of
equations above is (1, 1). Looking at
the graph to the right, what do you
notice about this point? What makes
it special?
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How can we check to see that the
point (1, 1) is truly a solution to the
system of equations?
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Example: Graph the following two lines (system of equations) on the graph provided:
y=x+1
y = -2x + 4
By looking at the example above and
using your graph, what is the solution
to the system of equations?
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How do you know?
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Note: When given the graphs of the two lines in a system of equations, the solution to the system is the
point (x, y) that is common to both of these lines. This point is called the point of intersection of the two
lines, and is where the two lines cross at a single point.
Example:
To the right is a graph of two lines
that never cross. These lines are
called ___________ lines. Since
these lines never cross, they have the
__________ slope.
Since parallel lines never cross, there
can be no point of intersection. That
is, for a system of equations that
graphs as parallel lines, there can be
no solution.
Example: Find the solution to the following system of equations by graphing the lines. Check your
solution by substituting the solution into both equations to make sure it works!
y = 3x – 3
y = -2x + 7