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Solving Systems of Equations by Elimination
Lesson 6.3 (pages 397403)
by
TweetRob Frederick
on 16 November 2011Transcript of Solving Systems of Equations by Elimination
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LINEAR EQUATIONS
BY
ELIMINATION
SOLVING SYSTEMS
OF
x + 2y = 6
x  y = 3
Bell Ringer
1.) Daniel's recipe for 24 cookies calls for 2 1/2 cups of flour. How much flour will Daniel need to make 60 cookies?
A.) 1 cup
B.) 6 1/4 cups
C.) 6 1/2 cups
D.) 7 1/2 cups
2.) Which graph shows a line where each value of y is three more than half of x?
A.) B.)
C.) D.)
REMEMBER...
two weeks ago we
graphed systems.
TODAY...
we will talk about elimination
as a third option.
run
SO...
NOW...
practice elimination.
Solve the system by elimination.
*
Write it down
*
CHECK...
*
ASSIGNMENT
Lesson 6.3
Worksheets
A
B
C
Practice
&
,
,
,
This is due on Friday...
We will work together on SOME
of it tomorrow in class...
*
*
*
*
*
*
*
*
Solve the system:
{
5x + 2y = 1
x  2y = 19
Notice that y has opposite coefficients in this system.
This is a clue to us that elimination is a viable option to solve the system.
5x + 2y = 1
x  2y = 19
SOLVING BY ELIMINATION
SOLVING BY ElIMINATION
Step 1: Align Like Terms
Step 2: Eliminate and Solve
SOLVING BY ELIMINATION
Step 3: Substitute and solve
SOLVING BY ELIMINATION
Step 4: Write the ordered pair and check
SOLVING BY ELIMINATION
Don't forget to check!!!
The four steps for solving with elimination.
_
_
_
_
_
1.) Write the system so that like terms are aligned
2.) Eliminate one of the variables and solve for the other variable.
3.) Substitute the value of the first variable into one of the original equations and solve for the second variable.
4.) Write the answers from steps 2 and 3 as an ordered pair, (x, y), and check.
OR...
Four steps: the short version (for your HW)...
1.) Align like terms
2.) Eliminate and solve
3.) Substitute and solve
4.) Write the ordered pair and check
{
3x + 4y = 18
2x + 4y = 8
Step 1:
Step 2:
Step 3:
Step 4:
3x + 4y = 18
2x + 4y = 8
3x + 4y = 18
2x + 4y = 8
(2, 3)
MORE PRACTICE...
Solve by elimination.
{
2x + y = 3
x + 3y = 12
Step 1: Align like terms
Which variable has coefficients that are the same?
What do we do if neither are the same?!
Step 1 (part 2)...
Step 2: Eliminate and Solve
Step 3: Substitute and Solve
After step 2 we should have gotten
y = 3
What do we do with this information?
Step 4: Rewrite and Check
After step 3 we now have the solution
(3, 3)
BUT DON'T FORGET TO CHECK YOUR WORK!
*
(
3
,
3
)
CHECK...
2
x
+
y
= 3
2(
3
) + (
3
) = 3
6 + (3) = 3
3 = 3

x
+ 3
y
= 12
(
3
) + 3(
3
) = 12
3 + (9) = 12
12 = 12
Checking your work takes time, BUT...
it means never having to get a solution wrong.
APPLICATION
Sam spent $24.75 to buy 12 flowers for his mother. The bouquet contained roses, which cost $2.50 each, and daisies, which cost $1.75 each. How many of each type of flower did Sam buy?
First,
We need to get two equations for the scenario.
Equation 1:
Equation 2:
2.50r + 1.75d = 24.75
r + d = 12
Let r be roses and d be daisies.
NOW...
Setup for elimination.
2.50r + 1.75d = 24.75
2.50(r + d = 12)
THEN...
Use elimation and solve.
SO...
Remember r was roses and d was daisies...
So Sam bought
5 roses and
7 daisies.
*
*
*
*
*
*
Then, last week we solved
using subsitution.
5x +
2y
= 1
x 
2y
= 19
While we could still solve by graphing or substitution, elimination may take fewer steps and so we will use it this time.
x's should be lined up
y's are lined up
...so are constants
x  2y = 19
5x + 2y = 1
_
_
_
6x = 18
6
6
_
_
x = 3
Since x = 3...
x  2y = 19
3  2y = 19
+3
+3
2y = 16
2
y = 8
_
_
(3, 8)
Remember this ordered pair is a solution for BOTH equations,
so when we check it must make BOTH of them true.
You might also remember that this is the point where the
graphs of these two equations intersect.
x
 2
y
= 19
5
x
+ 2
y
= 1
(
3
,
8
)
5(
3
) + 2(
8
) = 1
3
 2(
8
) = 19
15 + 16 = 1
3  16 = 19
1 = 1
19 = 19
If you take the time to check your work,
you NEVER have to get a problem wrong because of mistakes.
3x + 4y = 18
(2x + 4y = 8)
5x = 10
5 5
x = 2
2x + 4y = 8
2(2) + 4y = 8
4 + 4y = 8
+4 +4
4y = 12
4 4
y = 3
_
_
_
_
_
_
3x + 4y = 18
3(2) + 4(3) = 18
6 + 12 = 18
18 = 18
2x + 4y = 8
2(2) + 4(3) = 8
4 + 12 = 8
8 = 8
Sometimes we have to multiply one or both equations
before we can eliminate...
What should we multiply by this time?
2x + y = 3
x + 3y = 12
If we multiply the second equation by 2,
then the x's can eliminate each other.
2x + y = 3
2(x + 3y = 12)
2x + y = 3
2x + 6y = 24
Notice that nothing eliminates when
they are lined up.
2x + y = 3
2x + 6y = 24
Now, what eliminates?
Then can we solve?
What did you get for y?
Roses are $2.5o each
Daisies are $1.75 each
Cost was $24.75 total
There are 12 flowers total
Notice we multiplied equation 2 times 2.50.
This is so we can eliminate the r's.
We could have multiplied times 1.75 and eliminated the d's...
2.50r + 1.75d = 24.75
2.50r  2.50d = 30.00
2.50r + 1.75d = 24.75
2.50r  2.50d = 30.00
0.75d = 5.25
0.75 0.75
d = 7
r + d = 12
r + 7 = 12
7 7
r = 5
(5, 7)
Now we need to interpret the results.
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_
_
_
_
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_
_
_
_
Full transcript_
_
_
_
_
2
_
_
_
_
_
LINEAR EQUATIONS
BY
ELIMINATION
SOLVING SYSTEMS
OF
x + 2y = 6
x  y = 3
Bell Ringer
1.) Daniel's recipe for 24 cookies calls for 2 1/2 cups of flour. How much flour will Daniel need to make 60 cookies?
A.) 1 cup
B.) 6 1/4 cups
C.) 6 1/2 cups
D.) 7 1/2 cups
2.) Which graph shows a line where each value of y is three more than half of x?
A.) B.)
C.) D.)
REMEMBER...
two weeks ago we
graphed systems.
TODAY...
we will talk about elimination
as a third option.
run
SO...
NOW...
practice elimination.
Solve the system by elimination.
*
Write it down
*
CHECK...
*
ASSIGNMENT
Lesson 6.3
Worksheets
A
B
C
Practice
&
,
,
,
This is due on Friday...
We will work together on SOME
of it tomorrow in class...
*
*
*
*
*
*
*
*
Solve the system:
{
5x + 2y = 1
x  2y = 19
Notice that y has opposite coefficients in this system.
This is a clue to us that elimination is a viable option to solve the system.
5x + 2y = 1
x  2y = 19
SOLVING BY ELIMINATION
SOLVING BY ElIMINATION
Step 1: Align Like Terms
Step 2: Eliminate and Solve
SOLVING BY ELIMINATION
Step 3: Substitute and solve
SOLVING BY ELIMINATION
Step 4: Write the ordered pair and check
SOLVING BY ELIMINATION
Don't forget to check!!!
The four steps for solving with elimination.
_
_
_
_
_
1.) Write the system so that like terms are aligned
2.) Eliminate one of the variables and solve for the other variable.
3.) Substitute the value of the first variable into one of the original equations and solve for the second variable.
4.) Write the answers from steps 2 and 3 as an ordered pair, (x, y), and check.
OR...
Four steps: the short version (for your HW)...
1.) Align like terms
2.) Eliminate and solve
3.) Substitute and solve
4.) Write the ordered pair and check
{
3x + 4y = 18
2x + 4y = 8
Step 1:
Step 2:
Step 3:
Step 4:
3x + 4y = 18
2x + 4y = 8
3x + 4y = 18
2x + 4y = 8
(2, 3)
MORE PRACTICE...
Solve by elimination.
{
2x + y = 3
x + 3y = 12
Step 1: Align like terms
Which variable has coefficients that are the same?
What do we do if neither are the same?!
Step 1 (part 2)...
Step 2: Eliminate and Solve
Step 3: Substitute and Solve
After step 2 we should have gotten
y = 3
What do we do with this information?
Step 4: Rewrite and Check
After step 3 we now have the solution
(3, 3)
BUT DON'T FORGET TO CHECK YOUR WORK!
*
(
3
,
3
)
CHECK...
2
x
+
y
= 3
2(
3
) + (
3
) = 3
6 + (3) = 3
3 = 3

x
+ 3
y
= 12
(
3
) + 3(
3
) = 12
3 + (9) = 12
12 = 12
Checking your work takes time, BUT...
it means never having to get a solution wrong.
APPLICATION
Sam spent $24.75 to buy 12 flowers for his mother. The bouquet contained roses, which cost $2.50 each, and daisies, which cost $1.75 each. How many of each type of flower did Sam buy?
First,
We need to get two equations for the scenario.
Equation 1:
Equation 2:
2.50r + 1.75d = 24.75
r + d = 12
Let r be roses and d be daisies.
NOW...
Setup for elimination.
2.50r + 1.75d = 24.75
2.50(r + d = 12)
THEN...
Use elimation and solve.
SO...
Remember r was roses and d was daisies...
So Sam bought
5 roses and
7 daisies.
*
*
*
*
*
*
Then, last week we solved
using subsitution.
5x +
2y
= 1
x 
2y
= 19
While we could still solve by graphing or substitution, elimination may take fewer steps and so we will use it this time.
x's should be lined up
y's are lined up
...so are constants
x  2y = 19
5x + 2y = 1
_
_
_
6x = 18
6
6
_
_
x = 3
Since x = 3...
x  2y = 19
3  2y = 19
+3
+3
2y = 16
2
y = 8
_
_
(3, 8)
Remember this ordered pair is a solution for BOTH equations,
so when we check it must make BOTH of them true.
You might also remember that this is the point where the
graphs of these two equations intersect.
x
 2
y
= 19
5
x
+ 2
y
= 1
(
3
,
8
)
5(
3
) + 2(
8
) = 1
3
 2(
8
) = 19
15 + 16 = 1
3  16 = 19
1 = 1
19 = 19
If you take the time to check your work,
you NEVER have to get a problem wrong because of mistakes.
3x + 4y = 18
(2x + 4y = 8)
5x = 10
5 5
x = 2
2x + 4y = 8
2(2) + 4y = 8
4 + 4y = 8
+4 +4
4y = 12
4 4
y = 3
_
_
_
_
_
_
3x + 4y = 18
3(2) + 4(3) = 18
6 + 12 = 18
18 = 18
2x + 4y = 8
2(2) + 4(3) = 8
4 + 12 = 8
8 = 8
Sometimes we have to multiply one or both equations
before we can eliminate...
What should we multiply by this time?
2x + y = 3
x + 3y = 12
If we multiply the second equation by 2,
then the x's can eliminate each other.
2x + y = 3
2(x + 3y = 12)
2x + y = 3
2x + 6y = 24
Notice that nothing eliminates when
they are lined up.
2x + y = 3
2x + 6y = 24
Now, what eliminates?
Then can we solve?
What did you get for y?
Roses are $2.5o each
Daisies are $1.75 each
Cost was $24.75 total
There are 12 flowers total
Notice we multiplied equation 2 times 2.50.
This is so we can eliminate the r's.
We could have multiplied times 1.75 and eliminated the d's...
2.50r + 1.75d = 24.75
2.50r  2.50d = 30.00
2.50r + 1.75d = 24.75
2.50r  2.50d = 30.00
0.75d = 5.25
0.75 0.75
d = 7
r + d = 12
r + 7 = 12
7 7
r = 5
(5, 7)
Now we need to interpret the results.
_
_
_
_
_
_
_
_
_
_
_