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.

Solution: The one (1) point of intersection In this case: ( 5, 1 ) Called an INDEPENDENT (CONSISTENT) SYSTEM

Graph both equations below on the same graph.

Solution: Since there is no intersection point, no points satisfy both equations at the same time. “NO SOLUTION” Called an INCONSISTENT SYSTEM

Graph both equations below on the same graph.

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, 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

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: y = mx + b Dependent (and Consistent)