Objective - To identify the number of solutions
a system of linear equations may have.
S o lv e.
x 2y 6
1
x
x
y
x 1
2
2y
x 6
y
x 2y 6
2
2
1
y
x
2
m
b
1
1
1
y
x
2
m
2
1
b
x
1
2
3
No Solution
3
3 Possible Outcomes
1)
2)
Lines Intersect
One
Solution
3)
Lines Parallel
Lines Coincide
No
Solution
Infinitely
Many
Solutions
Solve .
y
1
3
2x
x
2x 6 y
12
2x
2x
6y
2x 12
2
6y
12
6
y
1
y
3
m
b
x
1
2
6
1
y
x
3
2
3
m
b
Infinitely Many Solutions
x
1
3
2
2
Solve the system graphically, by substitution, and
by elimination.
Graphic Method
x 2y 4
x y 3
x
2y
x
x
y
y
x
m
b
4
x
y
x
2y
2
3
x
3
x
1
x
4
2
1
y
2
x
1
m
2
3
1
2
,2
1
2
b
2
2
Solve the system graphically, by substitution, and
by elimination.
x
x
y
3
2y
4
x
x
y
3
x
y
y
y
y
x
Substitution
x
x
x
3
3
2
3
1
2
3
3
2
3
,2
1
3
2y
4
2 x 3
x 2x 6
6
3x
3
x
4
4
6
2
3
2
3
Solve the system graphically, by substitution, and
by elimination.
x
x
x
x
y
3
2y
4
y
2y
3
4
3y
3
7
3
y
7
3
Elimination
x
2y 4
7
x 2
4
3
14
x
4
3
12
x
3
2 1
,2
2
3
3
x
3
14
3
Solve the system graphically, by substitution, and
by elimination.
Graphic Method
4y 2 x 8
x 2y 3
4y
2x
x 2y
x
2y x
2
2
1
y
x
2
m
1
2
b
8
4
y
y
3
x
3
x
y
4
2
4
1
2
3
m
2
3
2
No Solution
b
8
x
4
x
2
1
2
2
Solve the system graphically, by substitution, and
by elimination.
x
2y
3
4y
2x
8
x
1
2y
2y
3
2y
x
2y
x
2y
x
2y
Substitution
4y
3
2x
8
4y
2 2y
3
4y
4y
3
4y
4y
4y 2
4y
0
2
3
6
8
8
False!
No Solution
Solve the system graphically, by substitution, and
by elimination.
x
2y
3
4y
2x
8
x
2y
3
4y
2x
8
Elimination
2( x
2x
2y
4y
3)
8
2x
4y
6
2x
4y
8
0
2
False!
No Solution