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UNIT 6 Solving Systems of Equations by Graphing You can solve a system of equations (2 or more equations) by graphing. Graph all equations on the same graph. The solution is the intersection point/s of the graphs. Graph the following system of equations (by hand) and state the solution/s if they exist. 1. y 3 x , and y x 2 2. 3 x y 2 , and y 3 x 5 (hint: put equations in slope-int. form) y y x x Solution/s: ____________________ Solution/s: ____________________ 3. 4 x 2 y 8 , and 12 x 6 y 24 4. y ( x 3) 2 1 , and y 3 y y x x Solution/s: ____________________ Solution/s: ____________________ 48 5. y 2 x , and y x y x Solution/s: ____________________ A graphing calculator can be used to find an approximate solution to a system of equations (the solution may be a decimal which has been rounded). Calculator Steps: 1. Enter equations in y1=, y2=, etc. and find an appropriate window where you can see all intersection points. 2. “calc”, “intersection” 3. Move curser close to the solution you are finding then answer: First curve?, enter, Second Curve?, enter, Guess?, enter 4. Repeat process if system contains additional solutions. Solve the following systems by graphing (calculator). 6. y 1 x 5 , and y ( x 2) 2 3 2 7. y 1.8 x 2 19 x 40 , and 5.4 x y 14 Solution/s: ____________________ Solution/s: ____________________ 8. (Page 363) The United States is researching a project called the National Aerospace Plane, which would regularly fly passengers into space. A model rocket launch was tested as part of this research project. 1 The model rocket flight can be modeled by the equation: h(t ) 88.2t (9.8)t 2 2 (where h is height in meters and t is time in seconds) How many seconds after take-off did the rocket reach a height of 200 feet? Hint: Graph the “model” equation and graph a line at 200 feet. Solution/s: ____________________ Assignment #12 worksheet Solving Systems by Graphing 49 3.2: Solving Systems of Equations Algebraically There are two different algebraic methods for solving a system of two equations. Substitution Method: Steps: 1. Solve one equation for a variable (a variable with a coefficient of 1 is easiest if this is possible). 2. Substitute for this variable the other equation. 3. You should only have one variable in this new equation. Solve for a value/s. 4. Use substitution to find the remaining part of the ordered pair/s. Solve the following systems using substitution. The systems may have 0, 1, 2, or infinite solutions. 1. 1 x y 7 2 4 x 2 y 16 8 x 4 y 5 2. y 2x 4 Solution/s: ____________________ Solution/s: ____________________ y x2 3. y x 6 y x 3 4. y 4 Solution/s: ____________________ Solution/s: ____________________ 50 Addition / Elimination Method Steps. 1. Algebraically put equations in the same “form”. 2. If necessary, multiply one or both equations by a constant so that when the equations are added together vertically one variable will “eliminate”. 3. Solve for the value of the remaining variable. 4. Use substitution to find the remaining part of the ordered pair for all solutions. Solve the following systems using Addition / Elimination. The systems may have 0, 1, 2, or infinite solutions. 3x y 7 1. 4 x 2 y 16 5 x 4 y 12 2. 6 y 7 x 40 Solution/s: ____________________ Solution/s: ____________________ y x 2 4 3. 2 y x 6 5m 2n 8 4. 4m 3n 2 Solution/s: ____________________ Solution/s: ____________________ Assignment #13 worksheet Solving Systems of Equations Algebraically 51 3.3, 4.9: Solving Systems of Inequalities by Graphing The intersection of the shaded regions is the solution of the system of inequalities. If the regions to not intersect, then no real solutions exist. Solve each system of inequalities by graphing. (The graph is a picture of the solution set.) y x 2 1. x 2 y 1 y 3 2. x 2 y y x x y 2( x 3) 2 3. 1 y x 5 3 y x 2 4 4. y x 1 y y x x Assignment #14 Complete the square where necessary: page 171 # 1, 3, 4 – 8, 11, 14, 15, 16, 44, 45; Page 304 # 20, 21, 23, 24, 79, 80, 83 Assignment #15 52 Solving Systems of Equations with Three Variables Systems of three equations can be solved with substitution or addition method. Graphically the solution point is a 3-dimensional point (x, y, z) Solve each system of equations. 1. Substitution Method 2. Substitution Method x 2 y z 9 3 y z 1 3z 12 x 2 y z 4 4 y 3z 1 y 5z 6 3. Addition / Elimination Method 2 x y 2 z 15 x y z 3 3x y 2 z 18 Assignment #16 Assignment #17 worksheet worksheet 4. Addition / Elimination Method 2 x y z 3 x 3 y 2 z 11 3 x 2 y 4 z 1 Unit 6 Review Unit 6 Test (with review) is next class – Be Prepared! 53