Download Substitution Method

Systems of Equations
Substitution Method
& Elimination Method
copyright © 2011 by Lynda Aguirre
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Substitution Method
Systems of Equations
2 Equations in 2 variables
copyright © 2011 by Lynda Aguirre
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Substitution Method
This method takes one equation and substitutes it into the other one.
Why are we doing this?
To get an equation with only one variable (unknown value) in it.
Step 1: Solve either equation for x or y (this will be the “original” equation)
Sometimes an equation
already has an equation that
is solved for x or y.
Call this equation
the “original”
Step 2: Replace x or y in the “other” equation with the value from the “original” equation.
Step 3: Solve for the remaining variable (in this case: x)
Now we have an
equation with only one
variable in it.
Add Like Terms and
Isolate x.
This gives us one variable
(x=4), now we need to find
the other one (y).
copyright © 2011 by Lynda Aguirre
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Substitution Method
Part II: Find the other variable
In steps 1-3, we plugged
the 2nd equation into the
1st and found x=4.
Step 4: Plug the value from step 3 (x=4) into the “original” equation
y  ( 4)  3
Step 5: Solve for the remaining variable (y).
y 1
This gives us both values
which we list as a
coordinate
Solution:
(4, 1)
copyright © 2011 by Lynda Aguirre
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Substitution Method
Step 1: Solve either equation for x or y (your choice)
1st equation is the
“original” equation
2nd equation (the “other” equation)
My choice: Solve the 1st equation for y:
Step 2: Replace the variable in the “other” equation with the value from the “original” equation
1st equation:
The
variable
The value of y
2nd equation:
Step 3: Solve for the remaining variable (in this case, solve for x)
Add Like Terms
and Isolate x.
This gives us one value
(4, ___)
Now we need to find the “y”
copyright © 2011 by Lynda Aguirre
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Substitution Method
Part II: Find the other variable
Step 4: Plug the value from step 3, (x=4), into the “original” equation
The original equation
Step 5: Solve for the remaining variable (y).
This gives us both values
which we list as a
coordinate
Solution:
(4, 2)
copyright © 2011 by Lynda Aguirre
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Substitution Method
Solution: (6, 0)
Try this one on your own
STEPS
1) Solve one equation for x or y, label it “original”
2) Plug “original” into the “other” equation
3) Solve for 1st variable
4) Plug 1st variable into the “original” equation
5) Solve for 2nd variable
6) Write the solution (x, y)
Note: if the problem has letters other than x and y in it,
put them in alphabetical order
copyright © 2011 by Lynda Aguirre
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Substitution Method: steps
STEPS
1) Solve one equation for x or y, label it “original”
2) Plug “original” into the “other” equation
3) Solve for 1st variable
4) Plug 1st variable into the “original” equation
5) Solve for 2nd variable
6) Write the solution (x, y)
Note: if the problem has letters other than x and y in it,
put them in alphabetical order
Things to note:
--In step 1, if you choose to solve for a variable with a coefficient, you will create fractions.
--You must substitute into one equation in step 2 and then the other one in step 4
--You can check your answers by plugging the numbers (x,y) into BOTH equations
--Sometimes step 1 is not necessary if one of the equations is already solved for x or y
copyright © 2011 by Lynda Aguirre
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copyright © 2011 by Lynda Aguirre
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Dependent and Inconsistent Systems of Equations
All the examples up to this point were systems of equations that (if
graphed) cross at a single point.
Lines that cross at a point (x, y) are
“Consistent Systems”.
But it is possible for two lines to be parallel (i.e. they never cross)
A system of parallel Lines is called an
“Inconsistent System”
OR Two lines could represent the same line graphed twice (i.e. one
on top of the other, so they intersect at every point)
The same line graphed twice is called a
“Dependent System”
copyright © 2011 by Lynda Aguirre
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Types of Systems and Solutions
Type of
System
Solution
Graph
Consistent
(x,y)
Two lines that
cross
Inconsistent
No solution
Parallel lines
Dependent
An Infinite
Number of
Solutions
Same line
twice (looks like
copyright © 2011 by Lynda Aguirre
one line)
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Solution: no solution
TOS: Inconsistent
Solution: An infinite number
of solutions
TOS: Dependent
Solution: no solution
TOS: Inconsistent
Solution: An infinite number
of solutions
TOS: Dependent
copyright © 2011 by Lynda Aguirre
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Elimination Method
Systems of Equations
2 Equations in 2 variables
copyright © 2011 by Lynda Aguirre
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Elimination (or Addition) Method
This method takes one equation and adds it to the other one.
Why are we doing this?
To get an equation with only one variable (unknown value) in it.
Sometimes one or both
equations are already in the
correct format
Step 2: Draw a line underneath and add the like terms (straight down). One should cancel out.
Now we have an
equation with only one
variable in it.
Step 3: Solve for the remaining variable (in this case: x)
Solution:
(4 , 2)
Step 4: Substitute this value into “either” of the original equations
copyright © 2011 by Lynda Aguirre
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Elimination (or Addition) Method
Step 1: Put both equations into General Form
Sometimes one or both
equations are already in the
correct format
Step 1a: If necessary, multiply one equation (or both) by a number and/or a
negative sign so x’s or y’s will cancel (i.e. equal zero)when added
My choice:
Now the x’s have
Make the
the same number
x’s cancel
and different signs
Step 2: Draw a line underneath and add the like terms (straight down). One should cancel out.
Now we have an equation with only one variable in it.
Step 3: Solve for the remaining variable (in this case: y)
Step 4: Substitute this value into “either” of the original equations
copyright © 2011 by Lynda Aguirre
Solution:
( -4, 7)
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Elimination (or Addition) Method
Try this one on your own
Solution: (-4, 7)
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary
Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable
Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre
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Elimination (or Addition) Method
Try this one on your own
Solution: (-4, 1)
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary
Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable
Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre
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Elimination (or Addition) Method
Solution: (-1, -6)
Try this one on your own
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary
Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable
Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre
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Elimination (or Addition) Method
Solution: (-8, -2)
Try this one on your own
ELIMINATION STEPS
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant if necessary
Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel
Step 3: Solve for the remaining variable
Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
copyright © 2011 by Lynda Aguirre
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Elimination Method: steps
Preparation:
Step 1: Put both equations in General Form:
Step 1a: Multiply one (or both) equations by a constant
Elimination Process:
Step 2: Draw a line underneath and add the like terms (straight down)
Step 3: Solve for the remaining variable
Step 4: Plug the value from step 3 into either of the original equations
Step 5: Solve for the remaining variable
Write the solution as a point: (x, y)
Things to note:
--You can check your answers by plugging the numbers (x,y) into BOTH equations
--Sometimes step 1 is not necessary if the equations are already in General Form
--If there are fractions in either equation, multiply by the LCD to get rid of them
--If there are decimals in either equation, multiply by a power of 10 (10, 100, 1000,…)
copyright © 2011 by Lynda Aguirre
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copyright © 2011 by Lynda Aguirre
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Using the Elimination Method, name the Solution and the Type of System (TOS)
Solution: no solution
TOS: Inconsistent
Solution: An infinite number
of solutions
TOS: Dependent
Solution: no solution
TOS: Inconsistent
Solution: An infinite number
of solutions
TOS: Dependent
copyright © 2011 by Lynda Aguirre
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