1) A system of linear equations is two or more linear equations whose solution is being sought. The solution is the ordered pair(s) that satisfy all equations. 2) There are three types of systems: consistent if lines intersect at one point (one solution), inconsistent if lines are parallel (no solution), and dependent if equations graph the same line (infinite solutions). 3) In the examples, (-4,16) is the solution to one consistent system while (-2,3) is not a solution to another since the equations are not equal at that point.

1. Systems of Linear Equations

2. Solving Systems of Equations Graphically

3. Definitions A system of linear equations is two or more linear equations whose solution we are trying to find. (1) y = 4 x – 6 (2) y = – 2 x A solution to a system of equations is the ordered pair or pairs that satisfy all equations in the system. The solution to the above system is (1, – 2).

4. Solutions Determine if ( – 4, 16) is a solution to the system of equations. y = – 4 x y = – 2 x + 8 (1) y = – 4 x 16 = – 4( – 4) 16 = 16 (2) y = – 2 x + 8 16 = – 2( – 4) + 8 16 = 8 + 8 16 = 16 Yes, it is a solution Example:

5. Solutions Determine if ( – 2, 3) is a solution to the system of equations. x + 2 y = 4 y = 3 x + 3 (1) x + 2 y = 4 – 2 + 2(3) = 4 – 2 + 6 = 4 4 = 4 (2) y = 3 x + 3 3 = 3( – 2) + 3 3 = – 6 + 3 3 = – 3 But… Example: So it is NOT a solution

6. Types of Systems The solution to a system of equations is the ordered pair (or pairs) common to all lines in the system when the system is graphed. ( – 4, 16) is the solution to the system. y = – 4 x y = – 2 x + 8

7. Types of Systems If the lines intersect in exactly one point, the system has exactly one solution and is called a consistent system of equations .

8. Types of Systems If the lines are parallel and do not intersect, the system has no solution and is called an inconsistent system . y = 6 x y = 6 x – 5 There is no solution because the lines are parallel.

9. Types of Systems If the two equations are actually the same and graph the same line, the system has an infinite number of solutions and is called a dependent system . y = 0.5 x + 4 x – 2 y = – 8 There is an infinite number of solutions because each equation graphs the same line.