3.1 Solving Linear Systems by
Graphing
10/1/12
Vocabulary
System of 2
Linear Equations:
A system consisting of two linear
equations in two variables.
Ex: 6x – 2y = 8
3x – y = 4
Solution of a system
of 2 linear equations:
Is an ordered pair (x, y) that
satisfies both equations.
Graphically, it’s the point where
the lines intersect.
Tell whether the ordered pair (3, 4) is a solution of
-2x + y = -2
4x – 2y = 3
Substitute 3 for x and 4 for y in BOTH equations.
-2(3) + 4 = -2
- 6 + 4 = -2
Answer: Not a Solution
4(3) – 2(4) = 3
12 – 8 = 3
Tell whether the ordered pair (3, 4) is a solution of
x + 2y = 11
2x – y = 2
Substitute 3 for x and 4 for y in BOTH equations.
3 + 2(4) = 11
3 + 8 = 11
2(3) – 4 = 2
6–4=2
Answer: Solution
Example 1
Solve a System by Graphing
Solve the system by graphing. Then check your solution.
y = –x + 3
y = 2x + 9
ANSWER
( –2, 5 )
You can check the solution by substituting -2 for x and
5 for y into the original equations.
y=-x+3
5= -(-2) + 3
5= 5
y=2x+9
5 = 2(-2) + 9
5 = -4 + 9
5=5
Example 2
Solve a System by Graphing
Solve the system by graphing. Then check your
solution algebraically.
3x – y = 3
In slope int. form: y = 3x - 3
x + 2y = 8
In slope int. form: y = -
ANSWER
1
x
2
+4
( 2, 3 )
Example 2
Solve a System by Graphing
You can check the solution by substituting 2 for x and 3
for y into the original equations.
Equation 1
Equation 2
3x – y = 3
x + 2y = 8
3( 2) – 3 =? 3
? 8
2 + 2( 3) =
? 3
6 – 3=
? 8
2 +6 =
3=3
8=8
ANSWER
The solution of the system is ( 2, 3 ).
Extra Example
Solve the system by graphing. Then check your solution.
1. x – 3y = 1
–x + y = – 1
ANSWER
( 1, 0 )
Solve a System by Graphing
Checkpoint
Solve the system by graphing. Then check your solution.
2. – x + 4y = 2
2x – 3y = 6
ANSWER
( 6, 2 )
Homework
WS 3.1
Number of Solutions
1 solution
: the lines have
different slopes
Infinitely many
solutions
:the lines have the
same equation.
No solution
:the lines are parallel
(same slope)
Example 3
Systems with Many or No Solutions
Tell how many solutions the linear system has.
a.
b. x + 2y = 4
2x – y = 1
x + 2y = 1
– 4x + 2y = – 2
2x  y  1
 2x
-2 x
 y  2 x  1
-1
-1 -1
y  2x  1
 4 x  2 y  2
 4x
 4x
x  2y  4
x
x  2y 1
x
-x
-x
2 y  4x  2
2 y  x  4
2 y  x 1
2
2
2
2 2
y  2x  1
Infinitely many solutions
:the lines have the same equation.
2
2
1
y x2
2
2
2
1
1
y x
2
2
No solution
:the lines are parallel (same slope)
Write and Use Linear Systems
Checkpoint
Tell how many solutions the linear system has.
3. 2x + 3y = 1
4x + 6y = 3
ANSWER 0
4. x – 4y = 5
– x + 4y = – 5
ANSWER infinitely many
solutions
5. x – 5y = 5
x + 5y = 5
ANSWER
1