Transcript LINEAR SYSTEMS – Substitution Method

LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 1 : Solve the system using the substitution method.
 x  2 y  12
y  2x  9
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 1 : Solve the system using the substitution method.
 x  2 y  12
y  2x  9
One equation already has one of the
variables isolated, in this case “ y “, so
we will substitute y’s expression into the
other equation where y appears…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 1 : Solve the system using the substitution method.
 x  2 y  12
y  2x  9
 x  22x  9  12
One equation already has one of the
variables isolated, in this case “ y “, so
we will substitute y’s expression into the
other equation where y appears…
Notice now we only have “x” in the
equation…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 1 : Solve the system using the substitution method.
 x  2 y  12
y  2x  9
 x  22 x  9  12
 x  4 x  18  12
3x  18  12
3x  6
x2
One equation already has one of the
variables isolated, in this case “ y “, so
we will substitute y’s expression into the
other equation where y appears…
Distribute, combine any like terms, and
solve for “x”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 1 : Solve the system using the substitution method.
 x  2 y  12
y  2x  9
 x  22 x  9  12
 x  4 x  18  12
3x  18  12
3x  6
x2
Substitute x = 2 into the 2nd
equation and evaluate “y”
y  2x  9
y  22   9
y  49
y  5
Answer is ( 2 , - 5 )
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 1 : Solve the system using the substitution method.
 x  2 y  12
y  2x  9
Answer is ( 2 , - 5 )
You should check your answer…
 x  2 y  12
 2  2 5  12
 2  10  12
 12  12
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
None of the variables are isolated.
We will have to choose one of the
equations and isolate either x or y.
Then substitute the result into the
other equation for the variable we
isolated.
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
I am going to look at the second
equation and solve for “y”. It has a
coefficient of ( -1 ) so it will be the
easiest.
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
2 x  y  1
 2x
  2x
 y  1  2 x
I am going to look at the second
equation and solve for “y”. It has a
coefficient of ( -1 ) so it will be the
easiest.
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
2 x  y  1
 2x
  2x
 y  1  2 x
I am going to look at the second
equation and solve for “y”. It has a
coefficient of ( -1 ) so it will be the
easiest.
 y 1 2x


1 1 1
y  1 2 x
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
2 x  y  1
 2x
  2x
 y  1  2 x
 y 1 2x


1 1 1
y  1 2 x
Now substitute for “y”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
5x  31  2x  5
2 x  y  1
 2x
  2x
 y  1  2 x
 y 1 2x


1 1 1
y  1 2 x
Now substitute for “y”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
5x  31  2x  5
5 x  3  6 x  5
 1x  3  5
 1x  2
x2
Solve for “x”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
5x  31  2x  5
5 x  3  6 x  5
 1x  3  5
 1x  2
x2
Now substitute “x” into
either equation and
solve for “y”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
5x  31  2x  5
5 x  3  6 x  5
 1x  3  5
 1x  2
x2
52   3 y  5
10  3 y  5
 10
 10
Now substitute “x” into
either equation and
solve for “y”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 2 : Solve the system using the substitution method.
5 x  3 y  5
2 x  y  1
5x  31  2x  5
5 x  3  6 x  5
 1x  3  5
 1x  2
x2
52  3 y  5
10  3 y  5
 10
 10
 3 y  15
y 5
Answer is ( 2 , 5 )
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
Again, none of the variables are
isolated. We will have to choose one
of the equations and isolate either x
or y. Then substitute the result into
the other equation for the variable
we isolated.
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
2 x  3 y  1
 3y   3y
2 x  1  3 y
Again, none of the variables are
isolated. We will have to choose one
of the equations and isolate either x
or y. Then substitute the result into
the other equation for the variable
we isolated.
2x 1 3y


2
2
2
1 3
x
 y
2 2
Yes, you might have to deal with fractions…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
1 3
x
 y
2 2
Now substitute this in for “x”
in the other equation…
 1 3 
4  y   15 y  5
 2 2 
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
 1 3 
4  y   15 y  5
 2 2 
1 3
x
 y
2 2
 2  6 y  15 y  5
 2  21y  5
21y  7
Now substitute this in for “x”
in the other equation…
1
y
3
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
 1 3 
4  y   15 y  5
 2 2 
 2  6 y  15 y  5
 2  21y  5
21y  7
1
y
3
Now substitute this in for “y” in
either equation and solve for “x”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
1
2 x  3   1
 3
2 x  1  1
 1  1
2x  0
x0
 1 3 
4  y   15 y  5
 2 2 
 2  6 y  15 y  5
 2  21y  5
21y  7
1
y
3
Now substitute this in for “y” in
either equation and solve for “x”…
LINEAR SYSTEMS – Substitution Method
When using the substitution method, we replace one variable in terms of
the other variable. It changes an equation from two unknowns into an
equation with one unknown.
You might have to solve one of the equations first for one of the variables
before you substitute.
EXAMPLE # 3 : Solve the system using the substitution method.
2 x  3 y  1
4 x  15 y  5
1
2 x  3   1
 3
2 x  1  1
 1  1
2x  0
x0
Answer is
1

0
,


3

 2  6 y  15 y  5
 2  21y  5
21y  7
1
y
3