Systems%20of%20Linear%20Equations

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Systems of Linear Equations

• Using a Graph to Solve

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What is a System of Linear Equations?
A system of linear equations is simply two or
more linear equations using the same variables.
We will only be dealing with systems of two
equations using two variables, x and y.
If the system of linear equations is going to
have a solution, then the solution will be an
ordered pair (x , y) where x and y make both
equations true at the same time.
If the lines are parallel, there will be no
solutions. If the lines are the same, there will
be an infinite number of solutions.
We will be working with the graphs of linear
systems and how to find their solutions
graphically.
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How to Use Graphs to Solve Linear Systems
Consider the following system
x y 1 x 2y 5
Using the graph to the right, we can see that any
of these ordered pairs will make the first
equation true since they lie on the line.
We can also see that any of these points will
make the second equation true.
However, there is ONE coordinate that makes both
true at the same time
The point where they intersect makes both
equations true at the same time.
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• If the lines cross once, there
• will be one solution.
• If the lines are parallel, there
• will be no solutions.
• If the lines are the same, there
• will be an infinite number of solutions.

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How to Use Graphs to Solve Linear Systems
Consider the following system
x y 1 x 2y 5
We must ALWAYS verify that your coordinates
actually satisfy both equations.
To do this, we substitute the coordinate (1 , 2)
into both equations.
x y 1 (1) (2) 1 ?
x 2y 5 (1) 2(2) 1 4 5 ?
Since (1 , 2) makes both equations true, then (1
, 2) is the solution to the system of linear
equations.
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Graphing to Solve a Linear System
Start with 3x 6y 15 Subtracting 3x from both
sides yields 6y 3x 15 Dividing everything by
6 gives us
While there are many different ways to graph
these equations, we will be using the slope -
intercept form.
Similarly, we can add 2x to both sides and then
divide everything by 3 in the second equation to
get
To put the equations in slope intercept form, we
must solve both equations for y.
Now, we must graph these two equations.
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Graphing to Solve a Linear System
Using the slope intercept form of these
equations, we can graph them carefully on graph
paper.
Start at the y - intercept, then use the slope.
Lastly, we need to verify our solution is
correct, by substituting (3 , 1).
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Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a
linear system using a graph.
Step 1 Put both equations in slope - intercept
form.
Solve both equations for y, so that each equation
looks like y mx b.
Step 2 Graph both equations on the same
coordinate plane.
Use the slope and y - intercept for each equation
in step 1. Be sure to use a ruler and graph
paper!
Step 3 Estimate where the graphs intersect.
This is the solution! LABEL the solution!
Step 4 Check to make sure your solution makes
both equations true.
Substitute the x and y values into both equations
to verify the point is a solution to both
equations.
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Graphing to Solve a Linear System
Let's do ONE moreSolve the following system of
equations by graphing.
2x 2y 3 x 4y -1
Step 1 Put both equations in slope - intercept
form.
Step 2 Graph both equations on the same
coordinate plane.
Step 3 Estimate where the graphs intersect.
LABEL the solution!
Step 4 Check to make sure your solution makes
both equations true.