Chapter 13 Section 2 Solutions of Systems of

Chapter 13 Section 2 Solutions of Systems of Equations

System of Equations Consistent At least one solution Independent Exactly one Solution Dependent Infinitely Many Solutions Inconsistent No Solution

The different possibilities for the graphs of two linear equations are summarized in the next three slides.

Intersecting Lines Description of Slopes and Graph Intercepts of Lines Number of Solutions Type of System Intersecting Lines 1 Solution Consistent and Independent Different Slopes

Same Line Description of Slopes and Graph Intercepts of Lines Number of Solutions Type of System Same Line Infinitely many Consistent and Dependent Same Slope, Same Intercept

Parallel Line Description of Slopes and Graph Intercepts of Lines Number of Solutions Type of System Parallel Line 0 Inconsistent Same Slope, Different Intercept

Example 1 State whether each system is consistent and independent, consistent and dependent, or inconsistent. y=x+3 y=x-1 The graphs appears to be parallel lines. Since they do not intersect, there is no solution. This system is inconsistent.

Example 2 State whether each system is consistent and independent, consistent and dependent, or inconsistent. (-3, 5) y = 1/3 x + 4 x = -3 The graphs appears to intersect at the point at ( -3, 5). Because there is one solution, this system of equations is consistent and independent.

Example 3 State whether each system is consistent and independent, consistent and dependent, or inconsistent. Each equation has the x + 1/2 y = 2 2 x + y = 4 same graph. Because any ordered pair on the graph will satisfy both equations, there are infinitely many solutions. The system is consistent and dependent.

Your Turn • State whether each system is consistent and independent, consistent and dependent, or inconsistent. y = -x + 2 y = 3 x - 2

Your Turn • State whether each system is consistent and independent, consistent and dependent, or inconsistent. x–y=4 3 x – 3 y = 12

Example 4 Directions Determine whether each system of equations has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it.

Example 4 y=x+2 y = -3 x – 6 y=x+2 (-2, 0) y = -3 x - 6 The graph appears to intersect at (-2, 0). Therefore, this system of equations has one solution, (-2, 0). Check that (-2, 0) is a solution to each Equation.

Checking Example 4 y=x+2 0 = -2 + 2 Replace x with -2 0=0 and y with 0. y = -3 x - 6 0 = -3(-2) - 6 Replace x with -2 0=6– 6 and y with 0. 0=0 Therefore, the solution of the system of equations is (-2, 0)

Example 5 2 x + y = 4 2 x + y = 6 Write each equation in slope intercept form and then graph them. y = -2 x + 4 y = -2 x + 6

Example 5 y = -2 x + 4 y = -2 x + 6 y = -2 x + 4 The graphs have the same slope and different y-intercepts. The system of Equations has no solution.

Your Turn Determine whether each system of equations has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it. y=x+3 y = -2 x + 3 One Solution, (0, 3)

Your Turn Determine whether each system of equations has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it. 2 x + y = 6 4 x + 2 y = 12 Infinitely Many Solutions

Your Turn Determine whether each system of equations has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it. 3 x – y = 3 3 x – y = 0 No Solution