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7. 5 Special Types of Linear Systems EXAMPLE 1 Show that the linear system

7. 5 Special Types of Linear Systems EXAMPLE 1 Show that the linear system has NO SOLUTION Method 1 GRAPHING Rewrite each equation in slope-intercept form, graph. 2 x + y = 5 2 x + y = 1 The lines are parallel and do not intersect. No solution.

7. 5 Special Types of Linear Systems EXAMPLE 1 Show that the linear system

7. 5 Special Types of Linear Systems EXAMPLE 1 Show that the linear system has NO SOLUTION Method 2 SUBSTITUTION Rewrite each equation in slope-intercept form, graph. y = -2 x + 5 y = -2 x + 1 The variables are eliminated and you are left with a false statement. No solution.

7. 5 Special Types of Linear Systems CHECKPOINT 1 Show that the linear system

7. 5 Special Types of Linear Systems CHECKPOINT 1 Show that the linear system has no solution. x + 3 y = 4 2 x + 6 y = 4

7. 5 Special Types of Linear Systems EXAMPLE 2 Show that the linear system

7. 5 Special Types of Linear Systems EXAMPLE 2 Show that the linear system has INFINITE SOLUTIONS Method 1 GRAPHING Rewrite each equation in slope-intercept form, graph. -2 x + y = 3 -4 x + 2 y = 6 You can see equations are she same line. Infinite solutions.

7. 5 Special Types of Linear Systems EXAMPLE 2 Show that the linear system

7. 5 Special Types of Linear Systems EXAMPLE 2 Show that the linear system has INFINITE SOLUTIONS Method 2 LINEAR COMBINATION Multiply equation 1 by 2. -2 x + y = 3 2(-2 x + y) = 2(3) -4 x + 2 y = 6 The two equations are identical. Infinite solutions.

7. 5 Special Types of Linear Systems CHECKPOINT 2 Show that the linear system

7. 5 Special Types of Linear Systems CHECKPOINT 2 Show that the linear system has infinitely many solutions. x – 2 y = 4 -x + 2 y = -4

7. 5 Special Types of Linear Systems SUMMARY Schurz certified Number of Solutions of

7. 5 Special Types of Linear Systems SUMMARY Schurz certified Number of Solutions of a Linear System LINES INTERSECT Exactly one solution. PARALLEL LINES No solution. LINES COINCIDE Infinitely many solutions.