What is a System of Linear Equations

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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.
We will be working with the graphs of linear
systems and how to find their solutions
graphically.
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Possible Types of Solutions
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How to Use Graphs to Solve Linear Systems
Consider the following system
x y 1 x 2y 5
The solution to this system is the ordered pair
or pairs which will make both equations true.
By graphing the lines we can find the solution to
the system, if it exist.
For this system 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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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
Let's summarize! There are 4 steps to solving a
linear system using a graph.
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Example
Decide whether the ordered pairs (1,1) (0, 3)
are solutions of the following system 2x y
3 x - 2y -1
Solution 1.) Check to see whether (1,1)
satisfies both equations Is 2(1) 1 3 True
? Is 1- 2(1) -1 True ?
2.) Check to see whether (0,3) satisfies both
equations Is 2(0) 3 3 True ? Is 0- 2(3)
-1 True ?
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Substitution Method for Solving Systems of
Equations

• Start with an example.
• Solve the following system of equations
• y 2x -x y 3

• The first equation is already solved for Y.
  Thus, we can rewrite the second equation follows
• -x 2x 3. Combining terms gives
• x 3.

• Since x3, we can find the y-value by using
  either equation.

Y 2(3)6, So solution is the ordered pair (3,6)
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Example Using Substitution Method
Solve the following system of equations using the
substitution method 2x 7y -12
x 2y 3
The variable x in the second equation has a
coefficient 1. So lets solve the second
equation for x as follows x3-2y.

Now, lets take x3-2y plug into the other
equation as follows
2(3-2y) 7y -12. Combining terms gives
6-4y 7y -12 6 3y -12 3y -18
y -6
We can find x by plugging back into x3-2y. So
x 3 2(-6) x 3 12 15
15, -6)
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Substitution Method for Solving Linear Systems
Step 1 Solve one equation for either
variable Step 2 Substitute for the solved
variable in the other equation.
This will result in an equation with one
variable. Step 3 Solve the equation that has
one variable Step 4 Substitute the solution
from step 3 into either one of
the original equations Step 5 Check your
solution.

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Use the Substitution Method to Solve Linear
System 3x y 4 6x 2y 12
No Solution

System 2x 5y 8 4x 10y 16
(x, y) 2x 5y 8
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Use the Substitution Method to Solve Linear
System 1/2x 1/3 y -1/3 1/2x
2y -7
Step 1 is to remove fractions from both equations

Solution (2, -4)
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Use the TI83 Calculator
The graphing calculator will only accept entries
that with y, so we need to solve both
equations for y.
So we have y x/2 7 and y -x/3
3 Enter both Graphs into the calculator
Hit GRAPH. The graphs appears to intersect off
the window. We need to redo the window of
viewing. Go to WINDOW and increase the size of
Xmax to 20 Hit GRAPH

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Use the TI83 Calculator (Cont.)
Use the INTERSECT option to find where the two
graphs intersect. 2nd TRACE (CALC) 5
intersectMove spider close to the
intersection.Hit ENTER 3 times. Answer  x
12  and y -1