Solving Systems of Equations

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Solving Systems of Equations 3 Approaches
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Ms. Nong Adapted from Mrs. N. Newmans PPT
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Method 1 Graphically
POSSIBLE ANSWER
Answer (x, y) or (x, y, z)
Method 2 Algebraically Using Addition and/or
Subtraction
Answer No Solution
Answer Identity
Method 3 Algebraically Using Substitution
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In order to solve a system of equations
graphically you typically begin by making sure
both equations are in Slope-Intercept form.
Where m is the slope and b is the y-intercept.
Examples y 3x- 4 y -2x 6
Slope is 3 and y-intercept is - 4.
Slope is -2 and y-intercept is 6.
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How to Use Graphs to solve Linear Systems.
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Looking at the System Graphs

• 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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Check by substitute answers to equations
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In order to solve a system of equations
algebraically using addition first you must be
sure that both equation are in the same
chronological order.
Example
Could be
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Now select which of the two variables you want to
eliminate. For the example below I decided to
remove x.
The reason I chose to eliminate x is because
they are the additive inverse of each other.
That means they will cancel when added together.
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Now add the two equations together.
Your total is therefore
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Now substitute the known value into either one of
the original equations. I decided to substitute 3
in for y in the second equation.
Now state your solution set always remembering to
do so in alphabetical order.
-1,3
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Lets suppose for a moment that the equations are
in the same sequential order. However, you
notice that neither coefficients are additive
inverses of the other.
Identify the least common multiple of the
coefficient you chose to eliminate. So, the LCM
of 2 and 3 in this example would be 6.
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Multiply one or both equations by their
respective multiples. Be sure to choose numbers
that will result in additive inverses.
becomes
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Now add the two equations together.
becomes
Therefore
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Now substitute the known value into either one of
the original equations.
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Now state your solution set always remembering to
do so in alphabetical order.
-3,3
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In order to solve a system equations
algebraically using substitution you must have
one variable isolated in one of the equations.
In other words you will need to solve for y in
terms of x or solve for x in terms of y.
In this example it has been done for you in the
first equation.
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Now lets suppose for a moment that you are given
a set of equations like this..
Choosing to isolate y in the first equation the
result is
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Now substitute what y equals into the second
equation.
becomes
Better know as Therefore
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Lets look at another Systems solve by Substitution
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y 4x 3x y -21
Step 5 Check the solution in both equations.
3x y -21 3(-3) (-12) -21 -9
(-12) -21 -21 -21
y 4x -12 4(-3) -12 -12
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This concludes my presentation on simultaneous
equations. Please feel free to view it again at
your leisure. http//www.sausd.us//Domain/492