3.1 Solve Linear Systems by Graphing Algebra II

Definition A system of two linear equations in two variables x and y, also called a linear system, consists of two equations that can be written in the following form: Ax + By = CEquation 1 Dx + Ey = FEquation 2

Definition A solution of a system of linear equations in two variables is an ordered pair (x, y) that satisfies each equation. Solutions correspond to points where the graphs of the equations in the system intersect.

Example 1 Graph the linear system and estimate the solution. Then check the solution algebraically. 5x – 2y = -10 2x – 4y = 12

5x – 2y = -10 2x – 4y = 12 It appears that the lines intersect at (-4, -5). Does it satisfy both equations? 5x – 2y = -10 5(-4) – 2(-5) = = = -10 2x – 4y = 12 2(-4) – 4(-5) = = = 12

Classifying Systems If a system has no solution, the system is inconsistent. A system that has at least one solution is consistent. –A consistent system that has exactly one solution is independent. –A consistent system that has infinitely many solutions is dependent.

Number of Solutions of a Linear System Lines intersect at one point; consistent and independent. Lines coincide; consistent and dependent. Lines are parallel; inconsistent.

Example 2 6x – 2y = 8  -2y = -6x + 8 y = 3x - 4 3x – y = 4  -y = -3x + 4 y = 3x - 4 Therefore, the graphs are the same line; the system is consistent and dependent.

Example 3 -4x + y = 5  y = 4x x + y = -2  y = 4x - 2 The slopes are equal. Therefore, the graphs are two parallel lines; the system is inconsistent.

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