Do Now 1/18/12 In your notebook, explain how you know if two equations contain one solution, no solutions, or infinitely many solutions. Provide an example of each

Infinitely many solutions No solution One solution

Objective practice identifying the number of solutions of a linear system

Section 7.5 “Solve Special Types of Linear Systems” consists of two or more linear equations in the same variables. Types of solutions: (1) a single point of intersection – intersecting lines (2) no solution – parallel lines (3) infinitely many solutions – when two equations represent the same line

+ “Solve Linear Systems by Elimination” Multiplying First!!” Eliminated x (2) 4x + 5y = 35 8x + 10y = 70 Equation 1 + x (-5) 15x - 10y = 45 -3x + 2y = -9 Equation 2 23x = 115 “Consistent Independent System” x = 5 4x + 5y = 35 Equation 1 Substitute value for x into either of the original equations 4(5) + 5y = 35 20 + 5y = 35 y = 3 4(5) + 5(3) = 35 35 = 35 The solution is the point (5,3). Substitute (5,3) into both equations to check. -3(5) + 2(3) = -9 -9 = -9

“Solve Linear Systems with No Solution” Eliminated Eliminated 3x + 2y = 10 Equation 1 _ + -3x + (-2y) = -2 3x + 2y = 2 Equation 2 This is a false statement, therefore the system has no solution. 0 = 8 “Inconsistent System” No Solution By looking at the graph, the lines are PARALLEL and therefore will never intersect.

“Solve Linear Systems with Infinitely Many Solutions” Equation 1 x – 2y = -4 Equation 2 y = ½x + 2 Use ‘Substitution’ because we know what y is equals. Equation 1 x – 2y = -4 x – 2(½x + 2) = -4 x – x – 4 = -4 This is a true statement, therefore the system has infinitely many solutions. -4 = -4 “Consistent Dependent System” Infinitely Many Solutions By looking at the graph, the lines are the SAME and therefore intersect at every point, INFINITELY!

How Do You Determine the Number of Solutions of a Linear System? First rewrite the equations in slope-intercept form. Then compare the slope and y-intercepts. y -intercept slope y = mx + b Number of Solutions Slopes and y-intercepts One solution Different slopes No solution Same slope Different y-intercepts Infinitely many solutions Same y-intercept