Download Systems of equations 2 or more equations on the same graph

Systems of equations
2 or more equations on the same graph
What is the solution to an equation
with one variable?
• All of the number(s) that make it true.
3x + 2 = 14
-2 -2
3x = 12
x=4
Only the number 4 makes it true
Substitute it in to check 3(4) + 2 = 14
12 + 2 = 14
What is the solution to an equation
that has 2 variables?
• All of the numbers that make it true
Solutions
• y = 3x + 4
(x, y)
• 1 = 3(-1) + 4
(-1, 1)
• 7 = 3(1) + 4
(1, 7)
• -1 = 3(-2) + 4
(-2, -1)
• 4 = 3(0) + 4
(0, 4)
• If I plot all the different
possible numbers that
make it true, what will
I get?
**Systems of equations
• 2 or more equations on the same
graph
**Solution to a system of equations
• All numbers that make both
equations true
• Solutions to a system are the
point(s) where the equations
intersect on the graph
**If the solution to a system of
equations is the point where the
lines intersect….what are the
possibilties?
Intersecting Lines
• different slopes
• Intersect at one
point
• Exactly one solution
**If the solution to a system of
equations is the point where the
lines intersect….what are the
possibilties?
Parallel Lines
• Same slope
• Different y
intercepts
• Never intersect
• No solutions
**If the solution to a system of
equations is the point where the
lines intersect….what are the
possibilties?
Collinear Lines
•
•
•
•
Same slope
Same y intercept
Same Line
Infinite # of
solutions
Solving a system of linear
equations
We are going to look at different strategies
for solving systems of equations.
1. Solving by Graphing
If the solution to a system of linear equations
is the point where the two lines intersect…
We could graph both lines, and find the point
that they intersect.
When do you solve by graphing?
• When both equations have y by itself
• When one equation has y by itself and the
other can be made that way easily.
For example:
1. y = 2x + 4
y = 4x – 5
2. y = -3x +2
y – 4x = 8
3. 2x + 3y = 12
4x + 2y = -6
yes
yes
no
Practice C
• Determine the number of solutions for
each system of equations
OR
• Determine the number of times the graphs
of the two equations intersect
1 solution – intersect once
no solutions – never intersect
infinite # of solutions – always intersect
Solving systems of equations
by elimination
Best used when equations
• are in standard form
• easily moved into standard form
Since we cannot solve an equation with
two variables:
We eliminate one variable by adding
opposite coefficients
Practice D
1.
-4x + 2y = 8
4x – 3y = -10
Which coefficients are opposites?
The x coefficients are opposites
– 4 and 4
1.
-4x + 2y = 8
(+) 4x – 3y = -10
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
– 1y = -2
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
– 1y = -2
– 1 –1
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
y= 2
1. Eliminate a variable by
adding.
Add when:
Coefficients are same
Signs are different
2. Solve for the variable
remaining
3. Substitute in one of
the original equations
and solve
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
y= 2
-4x + 2(2) = 8
1. Eliminate a variable by
adding.
Add when:
Coefficients are same
Signs are different
2. Solve for the variable
remaining
3. Substitute in one of
the original equations
and solve
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
y= 2
-4x + 2(2) = 8
-4x + 4 = 8
1. Eliminate a variable by
adding.
Add when:
Coefficients are same
Signs are different
2. Solve for the variable
remaining
3. Substitute in one of
the original equations
and solve
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
y= 2
-4x + 2(2) = 8
-4x + 4 = 8
-4x = 4
1. Eliminate a variable by
adding.
Add when:
Coefficients are same
Signs are different
2. Solve for the variable
remaining
3. Substitute in one of
the original equations
and solve
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
y= 2
-4x + 2(2) = 8
-4x + 4 = 8
-4x = 4
x = -1
1. Eliminate a variable by
adding.
Add when:
Coefficients are same
Signs are different
2. Solve for the variable
remaining
3. Substitute in one of
the original equations
and solve
-4x + 2y = 8
(+) 4x – 3y = -10
0x – 1y = -2
y= 2
-4x + 2(2) = 8
-4x + 4 = 8
-4x = 4
x = -1
1. Eliminate a variable by
adding.
Add when:
Coefficients are same
Signs are different
2. Solve for the variable
remaining
3. Substitute in one of
the original equations
and solve
solution: x = -1 and y = 2
these 2 lines intersect at the point (-1, 2)
Practice D
2.
3a + b = 5
2a + b = 10
1. Eliminate a
variable by adding
• Add when:
• Coefficients are
same
• Signs are
different
2. Solve for the
variable remaining
3. Substitute in one
of the original
equations and
solve
3a + b = 5
2a + b = 10
1. Eliminate a
variable by adding.
• Add same
coefficient and
different signs
• Note in this case
they have same
coefficient, but
have the same sign
2. Solve for the
variable remaining
3. Substitute in one of
the original
equations and
solve
– ( 3a + b = 5
2a + b = 10
-3a – b = -5
-a
= 5
a
= -5
2(-5) + b = 10
-10 +b = 10
b = 20
)
We need to make
one of the signs
negative by
multiplying one of
the equations by
negative 1
1. Mult 1 equation by
neg one and cross
it out.
2. Add the 2
remaining
3. Solve the eq
4. Substitute to find
the other
These two lines intersect at the point (-5, 20)
3.
2x + 3y = 4
-2x – 3y = -4
1. Eliminate a
variable by adding
or subtracting.
• Add same
coefficient and
different signs
• Sub same
coefficient and
same signs
2. Solve for the
variable remaining
3. Substitute in one
of the original
equations and
solve
3.
2x + 3y = 4
-2x – 3y = -4
0 +0=0
Is this True?
Does 0 = 0?
yes, it is true, their are
infinite solutions.
These are collinear lines
and always intersect
1. Eliminate a
variable by adding
• Add same
coefficient and
different signs
• Sub same
coefficient and
same signs
2. Solve for the
variable remaining
3. Substitute in one
of the original
equations and
solve
4.
3x + 2y = 12
3x + 2y = 8
1. Eliminate a
variable by adding
or subtracting.
• Add same
coefficient and
different signs
• Sub same
coefficient and
same signs
2. Solve for the
variable remaining
3. Substitute in one
of the original
equations and
solve
4. – (3x + 2y = 12 )
1. Eliminate a
variable by adding
or
subtracting.
3x + 2y = 8
• Add same
coefficient and
-3x – 2y = -12
different signs
0 + 0 = -4
• Multiply top one by
negative 1 so the
Is this True?
signs will be
different
Does 0 = -4?
2. Solve for the
No, it is false, There are no variable remaining
solutions to this system.
3. Substitute in one
These are parallel lines and of the original
equations and
never intersect
solve
Translating
1. Ticket sales include tickets at $5 for
children and $8.50 for adults were $1,500
$5C
Amt of $ spent on
Children’s tickets
+ $8.50A
= $1,500
Amt of $ spent on
Adults’s tickets
What quantities do I know?
Equals the total of
$1,500
Price of student and adult tickets
What quantities do we not know?
Is there a total?
# of student and adult tickets
These are our variables
A total of $1,500
Do you see the words is, was or were?
Would it still mean the same thing if you
replaced it with the word “equals
Yes,
Translating
2. There are chickens and pigs in a barn.
When you count all the legs you get 43
2C
Number of chicken
legs
+ 4P
= 43
Number of
pig legs
What quantities do I know?
Equals the
total of 43 legs
# of legs a chicken and pig have
What quantities do we not know? # of pigs (4), # of chickens(2)
These are our variables
Is there a total?
A total of 43 legs
Do you see the words is, was, were or anything that
could mean equals? Would it still mean the same thing
if you replaced it with the word “equals Yes, you get