2.2
Intro to Linear Systems of Equations
Linear a relationship whose graph is a line,
whose variables increase or
decrease at a constant rate of change
System of Equations a set of two or more equations
Let's review these equations:
y=x+1
slope (m)
rise
run
y-intercept
(b)
y = 2x
slope (m)
rise
run
y-intercept
(b)
y=x+2
slope (m)
rise
run
y-intercept
(b)
slope (m)
rise
run
y-intercept
(b)
y = -x + 3
Where else will the lines cross?
What does all that mean?????
y=x+1
2=1+1
y = -x + 3
(1,2)
x=1
y=2
2 = -1 + 3
Systems of equations
y=x+1
y = -x + 3
point of intersection
(1,2)
Solution to the systems of equations: x = 1 and y = 2
Solve by Substitution
y = -x + 2
y=x-4
-x + 2 = y = x - 4
-x + 2 = x - 4
2 = 2x - 4
6 = 2x
3=x
y=x-4
y=6-4
y=2
Solve by Graphing
y = -x + 2
Graph both equations.
y=x-4
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y=x-4
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1
2
3
4
5
x
y = -x + 2
Solve by Graphing
y = -x + 2
Graph both equations.
y=x-4
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y=x-4
y = -x + 2
-1 = 3 - 4
-1 = -3 + 2
-1 = -1
-1 = -1
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1
2
3
4
(3, -1)
5
x
Solve by Substitution
y = 2x - 3
y=x-1
2x - 3 = y = x - 1
x - 3 = -1
x=2
y=x-1
y=2-1
y=1
Solve by Graphing
y = 2x - 3
Graph both equations.
y=x-1
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y=x-1
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1
2
3
4
5
x
y = 2x - 3
Solve by Graphing
y = 2x - 3
Graph both equations.
y=x-1
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y=x-1
y = 2x - 3
1=2-1
1 = 2(2) - 3
1=1
1=4-3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(2, 1)
1
2
3
4
5
x
1=1
Solve by Substitution
y = 3x - 4
y = -3x + 2
3x - 4 = y = -3x + 2
3x - 4 = -3x +2
6x - 4 = 2
6x = 6
x=1
y = 3x - 4
y = 3(1) - 4
y = -1
Solve by Graphing
y = 3x - 4
Graph both equations.
y = -3x + 2
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y = -3x + 2
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1
2
3
(1, -1)
4
5
x
y = 3x - 4
-1 = -3(1) + 2
-1 = 3(1) - 4
-1 = -3 + 2
-1 = 3 - 4
-1 = -1
-1 = -1
Solve by Substitution
y=x-3
y=x+2
x-3 =y=x+2
x-3=x+2
0x - 3 = 2
-3 = 2
no solution
∅
Solve by Graphing
y=x-3
Graph both equations.
y=x+2
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y=x+2
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1
2
3
4
5
x
y=x-3
Solve by Graphing
y=x-3
Graph both equations.
y=x+2
Find the coordinates for the point of
intersection.
y
5
Check by substituting the
coordinates into both equations.
4
3
y=x+2
y=x-3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1
2
3
4
5
x
The lines are parallel
so...
There is NO intersection point
so...
There is NO solution!!
3 Possible Solutions
for Linear Systems of Equations
intersecting
lines
touch once
@ a coordinate point
(x, y)
one solution
different slope
parallel
lines
never touch
no solution
same slope, but
different y-intercept
coinciding
lines
touch everywhere
lines are overlapping
all real # are solutions
same slope, and
same y-intercept