Download 3.1 Graphing Systems of Equations

Let’s Warm Up!
1) Solve the system of equations by graphing:
2x + 3y = 12
2x – y = 4
Answer: (3, 2)
2) Find the slope-intercept form for the equation of a
line that passes through (0, 5) and is parallel to a
line whose equation is 4x – y = 3?
Answer: y=4x+5
3)Solve 3│x – 5│= 12
Answer: x= 1, 9
Let’s chat about finals
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
Wednesday Jan 22nd : 2, 4, 6
Thursday Jan 23rd : 1, 3, 5

Minimum Days

Final Review sheet

due DAY OF FINAL
Mini Quiz Time!


3 graphing Questions
Get out a pencil please.
8-2 Substitution
Objective: To use the substitution
method to solve systems of
equations.
Two Algebraic Methods:


Substitution Method
Elimination Method  will learn
about next 
RECALL…Three Types of Solutions:
 Intersection is Solution
One Solution
Different slope
Different y-intercept
“Intersect at one point”
No Solution
Same slope
Different y-intercept
“Run parallel  Never intersect”
Infinite Solutions
Same slope
Same y-intercept
“Same line  Intersect infinitely”
Substitution Method

Use the substitution method when:

one equation is set equal to a variable

y = 2x + 1
or
x = 3y - 2
Example 1

Instead of x = 2 we have:
x=y+2
These are all the same!
x + 2y = 11
(y + 2) + 2y = 11 x = 3 + 2
x=5
3y + 2 = 11
Answer: (5,3)
3y = 9
y=3
Try with a Mathlete
1)
2)
y = 3x
x + 2y = -21
y = 2x – 6
3x + 2y = 9
Answers:
1) (-3,-9)
2) (3,0)
Example 2

x + 4y = 1 Solve for x (because there
2x – 3y = -9 is no number in front of it)
First, solve for a variable
x = -4y + 1
x = -4(1) + 1
2(-4y + 1) – 3y = -9
x = -3
-8y + 2 – 3y = -9
-11y + 2 = -9
-11y = -11
y=1
Answer: (-3,1)
TOO
1)
2y = -3x
4x + y = 5
Answers:
1) (2,-3)
2) (13,30)
2) 2x – y = -4
-3x + y = -9
Special Cases
x + y = 16
2y = -2x + 2
x = 16 – y
2y = -2(16 – y) + 2
2y = -32 + 2y + 2
2y = -30 + 2y
0 = -30
False
NO SOLUTION
6x – 2y = -4
y = 3x + 2
6x – 2(3x + 2) = -4
6x – 6x – 4 = -4
-4 = -4
True
INFINITELY MANY
TOO for Homework
1) y = -x + 3, 2y + 2x = 4
2) x + y = 0, 3x + y = -8
3) y = 3x – 7, 3x – y = 7
Homework
Pg. 467 #17-32 left column
Pg. 467 #17-32 Left Column
Solve using substitution.
17.
20.
y = 3x - 8
y= 4-x
4c = 3d + 3
c = d -1
23.
c - 5d = 2
2c + d = 4
26.
(3,1)
29.
(6, 7)
(2,0)
32.
x - 3y = 0
3x + y = 7
8x + 6y = 44
x - 8y = -12
21 7
( , )
10 10
(4,2)
1
x = y + 3 Infinitely
2
2x - y = 6 Many
MORE Explanations
The following slides have more
examples and explanations of the
substitution method.
Examples: Use the substitution method to solve the system of equations.
the substitution method when a variable is
1) 2x + 3y = 2 (use
already isolated or when a variable has a
x – 3y = –17 coefficient of 1 and can easily be transformed)
1st: Transform
one equation
to isolate a
variable
2nd: Substitute into the
other equation and
solve for variable #1
3rd: Substitute into
transformed equation
from 1st step and solve
for variable #2
2x + 3y = 2 “x = 3y – 17”
x – 3y = –17
2(3y – 17) + 3y = 2
+3y +3y
6y – 34 + 3y = 2
x = 3y – 17
–34 + 9y = 2
+34
+34
(we picked
9y = 36
x – 3y = – 17
because x has a
9
9
coefficient of 1
y=4
x = 3y – 17 “y = 4”
x = 3(4) – 17
x = 12 – 17
x = –5
and can easily be
transformed)
Write answer as an ordered pair (x, y): One Solution
(–5 , 4)
Examples: Use the substitution method to solve the system of equations.
2) –9x + 3y = –21
3x – y = 7
1st: Transform
one equation
to isolate a
variable
3x – y = 7
-3x
-3x
–y = –3x + 7
-1
-1 -1
y = 3x – 7
(use the substitution method when a variable is
already isolated or when a variable has a
coefficient of 1 and can easily be transformed)
2nd: Substitute it into
the other equation and
solve for variable #1
3rd: Substitute into the
transformed equation
from 1st step and solve
for variable #2
–9x + 3y = –21 “y = 3x – 7”
–9x + 3(3x – 7) = –21
–9x + 9x – 21= –21
–21 = –21
True!!
(we picked
3x – y = 7 because
y has a coefficient
of -1 and can easily
be transformed)
Write answer as an ordered pair (x, y): Infinite Solutions
Examples: Use the substitution method to solve the system of equations.
3) 4x – 2y = 5
y = 2x + 1
1st: Transform
one equation
to isolate a
variable
y = 2x + 1
(already isolated)
(use the substitution method when a variable is
already isolated or when a variable has a
coefficient of 1 and can easily be transformed)
2nd: Substitute into the
other equation and
solve for variable #1
4x – 2y = 5
4x – 2(2x + 1) = 5
4x – 4x – 2 = 5
–2=5
3rd: Substitute into
transformed equation
from 1st step and solve
for variable #2
“y = 2x + 1”
False!!
No Solution
Chapter 8 Systems of Equations
8-2 Substitution
We will become experts at solving
systems of equations with
substitution.
Math Lab

Solve the system with substitution:
x=y+2
These are all the same!
x + 2y = 11
(y + 2) + 2y = 11 x = 3 + 2
x=5
3y + 2 = 11
Answer: (5,3)
3y = 9
y=3
Math Lab Review
Substitution
2)
x+y=0
3x + y = -8
1)
y = -x + 3
2y + 2x = 4
3)
y = 3x – 7
3x – y = 7
Special Cases
x + y = 16
2y = -2x + 2
x = 16 – y
2y = -2(16 – y) + 2
2y = -32 + 2y + 2
2y = -30 + 2y
0 = -30
False
NO SOLUTION
6x – 2y = -4
y = 3x + 2
6x – 2(3x + 2) = -4
6x – 6x – 4 = -4
-4 = -4
True
INFINITELY MANY
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