Chapter 2 Systems of Linear Equations and Matrices Section 2.1 Solutions of Linear Systems

Solutions of First - Degree Equations A solution of a first –degree equation in two unknowns is an ordered pair, and the graph of the equation is a straight line. A solution of a first –degree equation in two unknowns is an ordered pair, and the graph of the equation is a straight line.

Possible Solutions Unique solution Unique solution Inconsistent System Inconsistent System Dependent System Dependent System

Unique Solution When the graphs of two first-degree equations intersect, then we say the point of intersection is the solution of the system. When the graphs of two first-degree equations intersect, then we say the point of intersection is the solution of the system. This solution is unique in that it is the only point that the systems have in common. This solution is unique in that it is the only point that the systems have in common. The solution is given by the coordinates of the point of intersection. The solution is given by the coordinates of the point of intersection.

Inconsistent Systems When the graphs of two first-degree equations never intersect (in other words, they are parallel), there is no point of intersection. When the graphs of two first-degree equations never intersect (in other words, they are parallel), there is no point of intersection. Since there is not a point that is shared by the equations, then we say there is no solution. Since there is not a point that is shared by the equations, then we say there is no solution.

Dependent System When the graphs of two first-degree equations yield the exact same line, we say that the equations are dependent because any solution of one equation is also a solution of the other. When the graphs of two first-degree equations yield the exact same line, we say that the equations are dependent because any solution of one equation is also a solution of the other. Dependent systems have an infinite number of solutions. Dependent systems have an infinite number of solutions.

Solving Systems of Equations There are many methods by which a system of equation can be solved: There are many methods by which a system of equation can be solved: GraphingGraphing Echelon Method (using transformations)Echelon Method (using transformations) Substitution MethodSubstitution Method Elimination MethodElimination Method

Elimination Method Try to eliminate one of the variables by creating coefficients that are opposites. Try to eliminate one of the variables by creating coefficients that are opposites. One or both equations may be multiplied by some value in order to get opposite coefficients. One or both equations may be multiplied by some value in order to get opposite coefficients.

Example 1 Solve the system below and discuss the type of system and solution. Solve the system below and discuss the type of system and solution. x + 2y = 12 -3x – 2y = x – 2y = -18

Example 1 x + 2y = 12 x + 2y = 12 -3x – 2y = x = -6 -2x = -6 x = 3 x = 3 Solve for y:x + 2y = y = 12 2y = 9 y = 4.5 y = 4.5

Example 1 Check x = 3 and y = 4.5 in other equation. Check x = 3 and y = 4.5 in other equation. -3x – 2y = (3) – 2(4.5) = (3) – 2(4.5) = – 9 = – 9 = = -18 √ -18 = -18 √ Solution: Solution: Unique solution: (3, 4.5) Independent system

Example 2 Solve the system below and discuss the type of system and solution. Solve the system below and discuss the type of system and solution. 2x – y = 3 6x – 3y = 9

Example 3 Solve the system below and discuss the type of system and solution. Solve the system below and discuss the type of system and solution. x + 3y = 4 x + 3y = 4 -2x – 6y = 3 -2x – 6y = 3

Example 4 Solve the system below and discuss the type of system and solution. Solve the system below and discuss the type of system and solution. 4x + 3y = 1 3x + 2y = 2

Example 5 Solve the system below and discuss the type of system and solution. Solve the system below and discuss the type of system and solution.