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3.1 Solving Linear Systems by Graphing 10/1/12 Vocabulary System of 2 Linear Equations: A system consisting of two linear equations in two variables. Ex: 6x – 2y = 8 3x – y = 4 Solution of a system of 2 linear equations: Is an ordered pair (x, y) that satisfies both equations. Graphically, it’s the point where the lines intersect. Tell whether the ordered pair (3, 4) is a solution of -2x + y = -2 4x – 2y = 3 Substitute 3 for x and 4 for y in BOTH equations. -2(3) + 4 = -2 - 6 + 4 = -2 Answer: Not a Solution 4(3) – 2(4) = 3 12 – 8 = 3 Tell whether the ordered pair (3, 4) is a solution of x + 2y = 11 2x – y = 2 Substitute 3 for x and 4 for y in BOTH equations. 3 + 2(4) = 11 3 + 8 = 11 2(3) – 4 = 2 6–4=2 Answer: Solution Example 1 Solve a System by Graphing Solve the system by graphing. Then check your solution. y = –x + 3 y = 2x + 9 ANSWER ( –2, 5 ) You can check the solution by substituting -2 for x and 5 for y into the original equations. y=-x+3 5= -(-2) + 3 5= 5 y=2x+9 5 = 2(-2) + 9 5 = -4 + 9 5=5 Example 2 Solve a System by Graphing Solve the system by graphing. Then check your solution algebraically. 3x – y = 3 In slope int. form: y = 3x - 3 x + 2y = 8 In slope int. form: y = - ANSWER 1 x 2 +4 ( 2, 3 ) Example 2 Solve a System by Graphing You can check the solution by substituting 2 for x and 3 for y into the original equations. Equation 1 Equation 2 3x – y = 3 x + 2y = 8 3( 2) – 3 =? 3 ? 8 2 + 2( 3) = ? 3 6 – 3= ? 8 2 +6 = 3=3 8=8 ANSWER The solution of the system is ( 2, 3 ). Extra Example Solve the system by graphing. Then check your solution. 1. x – 3y = 1 –x + y = – 1 ANSWER ( 1, 0 ) Solve a System by Graphing Checkpoint Solve the system by graphing. Then check your solution. 2. – x + 4y = 2 2x – 3y = 6 ANSWER ( 6, 2 ) Homework WS 3.1 Number of Solutions 1 solution : the lines have different slopes Infinitely many solutions :the lines have the same equation. No solution :the lines are parallel (same slope) Example 3 Systems with Many or No Solutions Tell how many solutions the linear system has. a. b. x + 2y = 4 2x – y = 1 x + 2y = 1 – 4x + 2y = – 2 2x y 1 2x -2 x y 2 x 1 -1 -1 -1 y 2x 1 4 x 2 y 2 4x 4x x 2y 4 x x 2y 1 x -x -x 2 y 4x 2 2 y x 4 2 y x 1 2 2 2 2 2 y 2x 1 Infinitely many solutions :the lines have the same equation. 2 2 1 y x2 2 2 2 1 1 y x 2 2 No solution :the lines are parallel (same slope) Write and Use Linear Systems Checkpoint Tell how many solutions the linear system has. 3. 2x + 3y = 1 4x + 6y = 3 ANSWER 0 4. x – 4y = 5 – x + 4y = – 5 ANSWER infinitely many solutions 5. x – 5y = 5 x + 5y = 5 ANSWER 1