Determinants
Graphs of Parallel Lines
Graphs of the Same Line
Graphs of Intersection
Practice
Determinants
 What is the use of the determinant?
 The determinant tells us if there is no
solutions/infinitely many or exactly one solution.
 Given this system of equations, what is the
determinant?
 ax+by=e
cx+dy=f

The determinant is a*d – b*c
Determinants
 What is the determinants of the following systems of
linear equations?
 x= -9
-3x=4
 -8x+3y=-24
5x+2y=15
 4x-5y=-13
6x-7y=-5
x= -9
-3x=4
-8x+3y=-24
5x+2y=15
4x-5y=-13
6x-7y=-5
(-8)*2 – 3*5 = -31
4*(-7) – (-5)*6 = 58
1*0 – (-3)*0 = 0
Determinants
 Now what does the determinant number mean?
 IF the determinant (a*d – b*c) is ZERO

THEN the system of linear equations has no solution or it has
infinitely many.
 This means that the lines are either PARALLEL or they are the
SAME line.
 IF the determinant (a*d – b*c) is NON-ZERO

THEN the system of linear equations has exactly one solution.
 This means that the two lines intersect!
Parallel Lines
 Parallel lines will have a determinant of ZERO.
 Take this system of linear equations
 -6x+3y=9
-12x+6y=-6
 (-6)*6 – 3*(-12)= 0
 This system is graphed
to the right.
Same Line
 Systems of linear equations that are the same line have
a determinant of ZERO
 Take the system, for example.
 2x+3y=9
4x+6y=18
 2*6 – 3*4 = 0
 This system is graphed
to the right
Intersection
 Systems of linear equations that intersect have a
determinant that is NON-ZERO.
 Take this system, for example.
 4x+6y=-3
6x-2y=2
 4*(-2) – 6*6=-44
 This system is graphed
to the right.
Practice: How many solutions?
4x-5y=19
6x+6y=8
4*6 – (-5)*6=54
3x+6y=10
-2x+4y=3
3*4 – 6*(-2)=0
8x-3y=2
2x+7y=9
8*7 – (-3)*2=62
15x-5y=25
5x+20y=47
15*20-(-5)*5=325
Practice: How many solutions?
4x-5y=19
6x+6y=8
No Solutions/Infinitely Many
3x+6y=10
-2x+4y=3
8x-3y=2
2x+7y=9
15x-5y=25
5x+20y=47
Exactly One Solution