Transcript Document

Systems of Equations
OBJECTIVES
• To understand what a system of
equations is.
• Be able to solve a system of equations
from graphing the equations
• Determine whether the system has one
solution, no solution, or an infinite
amount of solutions.
• Be able to graph equations without
using a graphing calculator.
Defining a System of
Equations
• A grouping of 2 or more equations,
containing one or more variables.
x+y=2
2y = x + 2
2x + y = 5
y = 5x - 7
6x - y = 5
How do we “solve” a
system of equations???
• By finding the point where two or more
equations, intersect.
x+y=6
y = 2x
6
Point of intersection
4
2
1 2
How do we “solve” a
system of equations???
• By finding the point where two or more
equations, intersect.
x+y=6
y = 2x
6
(2,4)
4
2
1 2
ax + by = c
2x + 3y = 6
-2x
-2x
3y = 6 - 2x
3
3 3
2
y=2x
3
2
y=- x+2
3
ax + by = c
(Standard Form)
WE WANT THIS FORM!!!
y = mx + b
(Slope- Intercept)
Non-Unique Solutions
No Solution: • when lines of a graph are parallel
• since they do not intersect, there is no solution
• the slopes are the same and the
equation must be in slope-intercept form
Example of No Solution
2x+y=8
y=-2x-3
Remember when
graphing the
equations need to be
in slope intercept
form!!!
Non-Unique Solutions
Infinite Solutions: • a pair of equations that have the
same slope and y-intercept.
• also call a Dependent System
Example of Infinite Solutions
2x+4y=8
4x+8y=16
Unique Solutions
One Solution:
• the lines of two equations intersect
• also called an Independent System
Graphing Systems in
Slope-Intercept Form
• 2y + x = 8
y = 2x + 4
Steps:
1. Get both equations in slope-intercept form
2. Find the slope and y intercept of eq. 1 then graph
3. Find the slope and y intercept of eq. 2 then graph
4. Find point of intersection
Example 1: 2y+x=8 and y=2x+4
Equation 1
Equation 2
y = 2x + 4
(0,4)
m= 2/1
B=4
The solution
to the
system is
(0,4)
2y + x = 8
-x
-x
2y=
8 -1x
2
2
2
1
y x4
2
Notice the slopes and
y intercepts are
different so there will

only be one solution
m= -1/2
b=4
Example 2: y=-6x+8 and y+6x=8
Equation 1
Equation 2
y=-6x+8
y+6x=8
b=8
-6x
-6x
y=8-6x
b=8
m=-6(down)
1(right)
Infinite Solution
m=-6(down)
1(right)
Notice both
equations have the
same intercept and
slope. This means
all the points are
solutions.
Example 3: x-5y=10 and -5y=-x+40
Equation 2
Equation 1
-5y=-x+40
__ __ ___
-5 -5 -5
x-5y=10
-x
-x
-5y=10-1x
__ __ __
1
y  x 8
5
-5
b=-8
1
y  2  x
5
m=1/5
The slopes are the
same with different
intercepts so the
lines are parallel
-5 -5
b=-2

No Solution
m=1/5

You Try Examples
Determine whether the following equations have one, none, or
infinite solutions by graphing on the graph paper provided.
1)
1
1
x  y 1
3
8
8x  3y  24
2)
2
y=
x-1
3
y=3
ANS: One Solution
(6,3)
ANS: Infinite Solutions
3)
x + 2y = 6
x + 2y = 8
ANS: No Solution
Equation 1
1
1
x  y 1
3
8
Slope Intercept Form


8
y
x8
3
Equation 2
8x+3y=24
8
y
x8
3
Infinite
Solution
Equation1
x  2y  6
1
y x3
2
Equation 2

No Solution
x  2y  8
1
y x4
2
Notice the slopes are the same
and the y-intercepts are

different.
This means that the
lines are parallel so they will
never intersect
Equation 1
2
y  x 1
3
Equation 2

Solution (6,3)
y=3
Notice the
equations have
different slopes
and y-intercepts
so the lines will
have 1 solution
Algebraically determine if a point is a solution
We must always verify a proposed solution
algebraically. We propose (1,6) as a
solution, so now we plug it in to both
equations to see if it works:
y = 6x and
6 = 6(1) and
(6)= 6 and
y = 2x + 4,
6=2(1)+4,
6= 6.
Yes, (1,6) Satisfies both equations!
Algebraically determine
if a point is a solution
• Determine if (2,12) is a solution to the
system.
• y=6x
and
y=2x+4
• 12=6(2) and
12=2(2)+4
• 12=12 and
12=8
• Yes
No
• (2,12) is NOT a solution to the system