Download 3-2 Solving Systems of Equations Algebraically

3-2 Solving Systems
Algebraically
SWBAT:
1) Solve systems of linear equations by using substitution
2) Solve real world problems by using systems of linear equations
Solve the System by Graphing…
They intersect at (2,4),
and both have it as a
solution.
Wait a Minute…
y = 3x - 2
x + 2 = 3x - 2
2 = 2x - 2
4 = 2x
2=x
• Notice in the first
equation y = x + 2
• Substitute in x + 2 for y in
the second equation.
• Solve for x.
We found x! How about y?
• Well y = x + 2 and y = 3x – 2
• We know x = 2
• Substitute in x to either
equation and solve for y!
y=x+2
y=2+2
y=4
y = 3x – 2
y = 3(2) – 2
y=6–2
y=4
So the Solution is (2,4)
Steps for Substitution
1. Pick one equation, solve for one variable.
(Solve in terms of x or y)
2. Substitute that expression equal to the
variable into the other equation. Solve for
the opposite variable.
3. Sub the solutions into one of the original
equations and find the other solution.
4. Write your solution as an ordered pair.
Ex 1: Solve using Substitution
x + 2y = 8
x = 8- 2y
1
x - y =18
1 2
(8 - 2y) - y =18
2
4 - y - y =18
4 - 2y =18
-2y =14
y = -7
x = 8- 2y
x = 8- 2(-7)
x = 8+14
x = 22
So the Solution
is (22, -7)
Ex 1: Solve using Substitution
x + 4y = 26
x = 26 - 4y
x - 5y = -10
26 - 4y - 5y = -10
26 - 9y = -10
-9y = -36
y=4
x = 26 - 4y
x = 26 - 4(4)
x = 26 -16
x =10
So the Solution
is (10, 4)
Ex 2: Compare Values
Ex 3: Consistent and
Dependent Systems
6x - 2y = -4
y = 3x + 2
6x - 2y = -4
6x - 2(3x + 2) = -4
6x - 6x - 4 = -4
-4 = -4
• Since the variables eliminated and the end result
is a true statement…this system has infinite
solutions. It is Consistent and Dependent.
Ex 3:Inconsistent
6x - 2y = -4
y = 3x - 6
6x - 2y = -4
6x - 2(3x - 6) = -4
6x - 6x +12 = -4
12 ¹ -4
• Since the end result was not balanced, there
would be no solutions. (Parallel Lines – no points
of intersection) The system is Inconsistent…
Ex 4: Word Problems
Mr. Falcicchio spent 40 minutes icing 24 cupcakes.
It took him 1 min to ice a vanilla cupcake and 2
minutes to ice a chocolate cupcake. How many of
each cupcake was made? Use a System to solve.
x = #vanilla
1x + 2y = 40
x = 24 - y
y = #chocolate 1(24 - y) + 2y = 40
x = 24 -16
x + y = 24
24 + y = 40
x =8
1x + 2y = 40
y = 16
x + y = 24
x = 24 - y
8 vanilla and 16 chocolate
cupcakes
Ex 4b: Word Problems
Mr. Frew coaches the Swim Team. He has 3 times
as many boys as girls. He has 88 swimmers. How
many Boys and Girls are there?
b + g = 88
b + g = 88
b = 3g
3g + g = 88
4g = 88
b = 3g
b = 3(22)
b = 66
g = 22
So Mr. Frew has 66 boy and 22 girl swimmers
Solving Systems of
Equations using
Elimination
You will be able to solve systems of equations using previous
methods as well as using elimination to solve for a variable.
Elimination using Addition
• Sometimes adding two equations together will eliminate one
variable. Using this step is called elimination.
• Once we eliminate one variable, we can solve for the
remaining variable.
• We will then substitute for that variable into one of the
equations in the system, in order to solve for the remaining
variable
• In order to use elimination the equations must be set up in
Standard Form. (x and y on same side)
Elimination with Addition
3x - 5y = -16
2x + 5y = 31
Notice how the y variables are opposites…
Add the two Equations together.
3x - 5y = -16
(+)2x + 5y = 31
2x + 5y = 31
2(3) + 5y = 31
5x = 15
x=3
(3,5) is the solution
6 + 5y = 31
5y = 25
y=5
More Practice Problems
1.
x + y = -3
x–y=1
2. 3m – 2n = 13
m + 2n = 7
Example using Elimination with
Same Signs
5s + 2t = 6
9s + 2t = 22
Notice how the t variables are equivalent…
Subtract the two expressions
5s + 2t = 6
(-)9s + 2t = 22
-4s = -16
5s + 2t = 6
5(4) + 2t = 6
20 + 2t = 6
s=4
(4,-7) is the solution
2t = -14
t = -7
Ex 2: Subtraction w/ Addition?
3a + b = 5
2a + b = 10
2a + b = 10
-1(2a + b) = -1(10)
-2a - b = -10
• Notice the b variables have the exact same
coefficient.
• Multiply one whole equation by -1 to change
signs!
3a + b = 5
(+) - 2a - b = -10
a = -5
(-5, 20) is the Solution
3a + b = 5
3(-5) + b = 5
-15 + b = 5
b = 20
Ex 2B: Elimination
-5r + 3s = -35
-5r + 2s = -30
-1(-5r + 2s) = -1(-30)
5r - 2s = 30
-5r + 3s = -35
(+)5r - 2s = 30
s = -5
-5r + 3s = -35
-5r + 3(-5) = -35
-5r -15 = -35
(4,-5) is the Solution
-5r = -20
r=4
What if the Variables Don’t Match?
• What would we do if our system of equations did
not have two variables with the same
coefficient?
• Ex: 3x + 4y = 6
5x + 2y = -4
• Can elimination still be used in order to solve the
system of equations?
Remember Multiplying by -1?
3x + 4y = 6
5x + 2y = -4
• We don’t always have
to multiply equations
by the same value.
• Notice how the ycoefficients are
multiples of 2.
• Multiply the bottom
equation by -2. What
happens? Can we use
elimination? Explain…
Ex 3: Solving Using Elimination
3x + 4y = 6
5x + 2y = -4
3x + 4y = 6
-10x - 4y = 8
-7x = 14
x = -2
5x + 2y = -4
-2(5x + 2y) = -2(-4)
-10x - 4y = 8
3x + 4y = 6
3(-2) + 4y = 6
-6 + 4y = 6
4y = 12
y=3
Determine the Best Method for
Solving the System of Equations
9x – 8y = 42
4x + 8y = -16
4x – 2y = 14
y=x
6x – y = 9
6x – y = 11
1/2x – 2/3y = 7/3
3/2 x + 2y = -25
Word Problem
• Find two numbers whose sum is 64 and whose difference is 42
Word Problem
• A youth group and their leaders visited Mammoth Cave. Two
adults and 5 students in one van paid 77 dollars. Two adults
and 7 students paid 95 dollars for the same tour. Find the
adult and student prices.