Graphing 2x2 Systems
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems etting Started c o n s i d e r the s y s t e m : y = -2x + 3
y = - 2x + 3 X
F o r e a c h equation, m a k e a table of ordered pairs (x,y). T h e n graph e a c h line. • W h i c h ordered pair a p p e a r s in both tables?
y =
/
i
X
0
0
1
1
2
2
3
3
4
4
2
X -
i
:
"
i
i
!1
;2
II
14
>
i
i
i
y
i 1
'
^
M
0
i
'\ 1,5
[S
-t
-3 -4 .at
•
W h a t do w e call this point, in terms of the s y s t e m ?
•
W h a t do w e call this point in terms of the g r a p h ?
1
1
::
IMPORTANT: W h e n the equations in a s y s t e m c a n be then the using the •
,
to the s y s t e m c a n be d e s c r i b e d of the
of
Why?
U s e the s l o p e and intercept of e a c h linear equation to generate the graphs. T h e n state the solution to the s y s t e m . 1)
System:
y=fx+2 y = 3x-1
-8
I
:
:
i
f
•
"
•
-6
-4 i
;
-
2) '
System:
^ y=x-3
r"
J
Solution:
©2012, T E S C C C
.^-s.
Solution:
07/18/12
page 1 of 4
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems C o n t i n u e graphing to find the solution to e a c h s y s t e m . y =- f
y =fx +7 3)
4) '
System: y = fx
x+1
System: y=^x+5 i
~—f \ i
1
6
1
3
6
4
I-S
••—
i
i 10
i 1
_4
—
-J.
1
t J
•
• 10
! j—
8
-4
"
i
^'
*.
-2
4
'
"i
6
:
;
8 [
Id'
.-2 -
1
-4
-S
! '•"-7-
Solution:
Solution:
y = 5 5)
System:
•10 f
3
.
S
4
6)
1-2
8
10
4
Solution:
©2012, T E S C C C
^ ^ y = 2x- 1 System: ^ ^ / = 3 + 2x
6
Solution:
07/18/12
page 2 of 4
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems ""ystems of equations c a n a l s o b e s o l v e d using the C A L C 5: intersect c o m m a n d o n a graphing calculator. H o w e v e r , before equations c a n be entered, y o u must solve e a c h equation for "y". System
Q.
- X + 2y = 9
E
3x + 4y = 8
CD CO
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Yi = 0 . 5 x + 4 . 5
Y2 = ( - f ) x + 2 K=-2
7)
System
E q u a t i o n s in "Y=" form
V=3.5:
S k e t c h the G r a p h
Table Check X
Y1
Y2
-4 -3 -2 -1 0
2.5 3 3.5 4 4.5
5 4.25 3.5 2.75 2
Table Check Y1
X
2x + y = 5
Solution
(-2, 3.5)
Solution
Y2
Yi =
4x + y = 12 Y2 =
8)
System
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Table C h e c k Y1
X
4x + 7y = 14 8y + 8 = 5x
Solution
Y2
Yi =
Y2 =
9)
System
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Table Check X
3x - 5y = 30
Y1
Solution
Y2
Yi =
6x + 20 = 10y Y2 =
10)
System
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Table Check X
-3x + y = 5 10 + 6x = 2y
Y1
Solution
Y2
Yi =
Y2 =
©2012, T E S C C C
07/18/12
page 3 of 4
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems
/
W h e n solving s y s t e m s (such a s by graphing), three different c a s e s c a n occur. T h e lines are
T h e lines
T h e lines
W h a t type of l i n e s ?
W h a t tvpe of s y s t e m ?
W h a t tvoe of s o l u t i o n s ?
solution (represented by the
solutions
An of solutions (or, every on the line).
)• S o m e t i m e s , the
Other V o c a b u l a r y
If the two equations in If the two equations in the s y s t e m graph into the s y s t e m h a v e the lines w h i c h intersect at is u s e d to d e s c r i b e the s a m e then they a single point, then they solution to s u c h a are called are called system. Symbol:
equations.
equations.
O n graph p a p e r or o n a calculator, s o l v e e a c h s y s t e m graphically. T h e n d e s c r i b e : • T h e type of l i n e s — p a r a l l e l , intersecting, or coincident (coinciding). T h e type of s y s t e m — inconsistent, consistent/independent, or consistent/dependent. T h e type of s o l u t i o n s — o n e solution, no solutions, or infinitely m a n y solutions. If the solution exists, state it a s a n ordered pair. y = 2x + 4 12)
y = - f x +1 y = -3x + 7
14)
y = -3x + 5 12x + 4 y = 2 0
©2012, T E S C C C
15)
4 x - 2y = - 1 0 - 6 x + 3 y = 12
07/18/12
y=|x-1 13) y = | x - 3
16)
y = 0.8x + 2 5x - 4y = 8
page 4 of 4
Graphing 2x2 Systems etting Started c o n s i d e r the s y s t e m : y = -2x + 3
y = - 2x + 3 X
F o r e a c h equation, m a k e a table of ordered pairs (x,y). T h e n graph e a c h line. • W h i c h ordered pair a p p e a r s in both tables?
y =
/
i
X
0
0
1
1
2
2
3
3
4
4
2
X -
i
:
"
i
i
!1
;2
II
14
>
i
i
i
y
i 1
'
^
M
0
i
'\ 1,5
[S
-t
-3 -4 .at
•
W h a t do w e call this point, in terms of the s y s t e m ?
•
W h a t do w e call this point in terms of the g r a p h ?
1
1
::
IMPORTANT: W h e n the equations in a s y s t e m c a n be then the using the •
,
to the s y s t e m c a n be d e s c r i b e d of the
of
Why?
U s e the s l o p e and intercept of e a c h linear equation to generate the graphs. T h e n state the solution to the s y s t e m . 1)
System:
y=fx+2 y = 3x-1
-8
I
:
:
i
f
•
"
•
-6
-4 i
;
-
2) '
System:
^ y=x-3
r"
J
Solution:
©2012, T E S C C C
.^-s.
Solution:
07/18/12
page 1 of 4
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems C o n t i n u e graphing to find the solution to e a c h s y s t e m . y =- f
y =fx +7 3)
4) '
System: y = fx
x+1
System: y=^x+5 i
~—f \ i
1
6
1
3
6
4
I-S
••—
i
i 10
i 1
_4
—
-J.
1
t J
•
• 10
! j—
8
-4
"
i
^'
*.
-2
4
'
"i
6
:
;
8 [
Id'
.-2 -
1
-4
-S
! '•"-7-
Solution:
Solution:
y = 5 5)
System:
•10 f
3
.
S
4
6)
1-2
8
10
4
Solution:
©2012, T E S C C C
^ ^ y = 2x- 1 System: ^ ^ / = 3 + 2x
6
Solution:
07/18/12
page 2 of 4
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems ""ystems of equations c a n a l s o b e s o l v e d using the C A L C 5: intersect c o m m a n d o n a graphing calculator. H o w e v e r , before equations c a n be entered, y o u must solve e a c h equation for "y". System
Q.
- X + 2y = 9
E
3x + 4y = 8
CD CO
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Yi = 0 . 5 x + 4 . 5
Y2 = ( - f ) x + 2 K=-2
7)
System
E q u a t i o n s in "Y=" form
V=3.5:
S k e t c h the G r a p h
Table Check X
Y1
Y2
-4 -3 -2 -1 0
2.5 3 3.5 4 4.5
5 4.25 3.5 2.75 2
Table Check Y1
X
2x + y = 5
Solution
(-2, 3.5)
Solution
Y2
Yi =
4x + y = 12 Y2 =
8)
System
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Table C h e c k Y1
X
4x + 7y = 14 8y + 8 = 5x
Solution
Y2
Yi =
Y2 =
9)
System
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Table Check X
3x - 5y = 30
Y1
Solution
Y2
Yi =
6x + 20 = 10y Y2 =
10)
System
E q u a t i o n s in "Y=" form
S k e t c h the G r a p h
Table Check X
-3x + y = 5 10 + 6x = 2y
Y1
Solution
Y2
Yi =
Y2 =
©2012, T E S C C C
07/18/12
page 3 of 4
Algebra 2 HS Mathematics Unit: 04 Lesson: 01
Graphing 2x2 Systems
/
W h e n solving s y s t e m s (such a s by graphing), three different c a s e s c a n occur. T h e lines are
T h e lines
T h e lines
W h a t type of l i n e s ?
W h a t tvpe of s y s t e m ?
W h a t tvoe of s o l u t i o n s ?
solution (represented by the
solutions
An of solutions (or, every on the line).
)• S o m e t i m e s , the
Other V o c a b u l a r y
If the two equations in If the two equations in the s y s t e m graph into the s y s t e m h a v e the lines w h i c h intersect at is u s e d to d e s c r i b e the s a m e then they a single point, then they solution to s u c h a are called are called system. Symbol:
equations.
equations.
O n graph p a p e r or o n a calculator, s o l v e e a c h s y s t e m graphically. T h e n d e s c r i b e : • T h e type of l i n e s — p a r a l l e l , intersecting, or coincident (coinciding). T h e type of s y s t e m — inconsistent, consistent/independent, or consistent/dependent. T h e type of s o l u t i o n s — o n e solution, no solutions, or infinitely m a n y solutions. If the solution exists, state it a s a n ordered pair. y = 2x + 4 12)
y = - f x +1 y = -3x + 7
14)
y = -3x + 5 12x + 4 y = 2 0
©2012, T E S C C C
15)
4 x - 2y = - 1 0 - 6 x + 3 y = 12
07/18/12
y=|x-1 13) y = | x - 3
16)
y = 0.8x + 2 5x - 4y = 8
page 4 of 4
