Download Honors Algebra 2/Trig

Chapter 3 – Linear Systems
3-1 Solving Systems
Using Tables and Graphs
 System of Linear Equations
 two or more linear equations using the same variables
 graphed on the same coordinate plane
 the solution is the point or set of points that makes all of
the equations true
 we will primarily look at two variable/two equation
systems
 We say that we find:
 The SOLUTION of the SYSTEM
 The SIMULTANEOUS SOLUTION of the EQUATIONS
(both of these describe the same “answer”)
Graph both equations below on the same graph.
x  y  6

x  y  4
x y 6
y  x  6
6
4
2
-10
-5
5
-2
-4
x y 4
 y  x  4
y  x4
-6
-8
10
 Solution:
x  y  6

x  y  4
 The one (1) point of intersection
 In this case: ( 5 , 1 )
 Called an INDEPENDENT
(CONSISTENT) SYSTEM
6
y  x4
4
2
-10
-5
5
10
-2
y  x  6
-4
-6
-8
Graph both equations below on the same graph.
3x  2 y  6


3
 y   2 x  1
3x  2 y  6
2 y  3x  6
3
y   x3
2
3
y   x 1
2
3x  2 y  6


3
 y   2 x  1
 Solution:
6
4
2
-10
-5
 Since there is no intersection point,
no points satisfy both equations at
the same time.
 “NO SOLUTION”
 Called an INCONSISTENT SYSTEM
5
10
-2
-4
-6
3
y   x 1
2
-8
3
y   x3
2
Graph both equations below on the same graph.
1
 x  2y  4
2

4 y  x  8  0
1
x  2y  4
2
1
2 y   x  4
2
1
y  x2
4
4y  x 8  0
4y  x 8
1
y  x2
4
6
1
 x  2y  4
2

4 y  x  8  0
4
y
2
-10
-5
1
x2
4
5
-2
-4
 Solution:
 EVERY point on the line is an intersection point
 EVERY point on the line is a solution
 There are an infinite number of solutions,
-6
-8
but the solution is NOT “ALL REAL NUMBERS”
 The solution is the line – write your solution as the equation of the line!
 {(x,y): y = ¼ x – 2}
 Called a DEPENDENT (CONSISTENT) SYSTEM
10
 Intersecting Lines
 One Solution that is a single point ( x , y )
 Independent (and Consistent)
 Parallel Lines
 No Solution
 Inconsistent
 Overlapping Lines
 Infinite number of solutions, all points on the line
 Express answer as the equation of the line:
 Dependent (and Consistent)
y = mx + b