5.2.3 Systems of Equations (Substitution Method)

Transcription

1 Lesson: Systems of Equations (Substitution Method) Alternate Systems of Equations (Substitution Method) Teacher Lesson Plan CC Standards 8.EE.8b 8.EE.8c Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Objective The student will determine the number of solutions of a system by analyzing a graph or by comparing the slopes and y-intercepts. The student will solve systems using the substitution method. Mathematical Practices #1 Make sense of problems and persevere in solving them. #5 Use appropriate tools strategically. #6 Attend to precision. #7 Look for and make use of structure. Teacher Input Bellwork: Review bellwork. Homework: Review important problems assigned the previous night. Introduction: Introduce as directed on the PowerPoint. Lesson: Teach as directed in the PowerPoint. Be sure to look at the notes on each slide for additional instruction and answers. Practice Pages 4-6 Closure Close as instructed in the PowerPoint. Be sure to look at the notes section of each slide. 1 P a g e

2 SECTION 1: Systems of Equations (Substitution Method) Number of Solutons Student Notes One Solution No Solution Infinitely Many Solutions The lines intersect. The equations will have: different slopes y = 3x + 6 y = 2x 4 The lines are parallel. The equations will have: same slope different y-intercepts y = 2x + 1 y = 2x 4 The lines coincide.** The equations will have: same slope same y-intercepts y = 2x 3 y = 2x 3 ** They are the same line! Guided Practice Without graphing, what will each system below look like? (intersecting lines, parallel lines, or same line) How many solutions will there be? (one solution, no solution, or infinitely many solutions) 1) y = x + 2 2) y = 4x + 7 3) 2x + 2y = 8 y = x + 1 y = 4x 1 y = x 4 You Try 1) y = x + 5 2) 3x + 2y = 5 4x + 4y = 12 3x + 2y = 6 2 P a g e

3 SECTION 2: Solving Systems using the Substitution Method NOTE Step 1 is not necessary because the first equation is already solved for y. Guided Practice 1) 2x + 7y = 3 2) 6x 2y = 4 3) x = 1 4y y = 3x + 2 x + 5y = 3 3x 2y = 8 You Try 1) y = 3x 2) 2x y = 8 3) x + 2y = 21 y = 2x 3 x y = 8 x + 3y = 12 4) A total of 75 brownies (b) and cakes (c) were donated for a bake sale to raise money for the football team. There were four times as many brownies donated as cakes. Part A) Part B) Write a system of equations to represent this situation. Solve to determine how many of each there were. 3 P a g e

4 Independent Practice Name Date Period: 4 P a g e

5 Independent Practice Name Date Period: Solve using the substitution method. 5 P a g e

6 19. Ms. Corley wants to take her class on a trip to either the nature center or the zoo. The nature center charges $4 per student plus $95 for a 1-hour naturalist program. The zoo charges $9 per student plus $75 for a 1-hour guided tour. a) Write a sysem of equations to represent this situation. b) At what number of students will the total nature center cost be the same as the zoo s? Solve your system using the substitution method. c) Ms. Corley has 22 students in her class. Determine which would cost less, the nature center or the zoo. 6 P a g e

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8 SECTION 1: Systems of Equations (Substitution Method) Number of Solutons Student Notes One Solution No Solution Infinitely Many Solutions The lines intersect. The equations will have: different slopes y = 3x + 6 y = 2x 4 The lines are parallel. The equations will have: same slope different y-intercepts y = 2x + 1 y = 2x 4 The lines coincide.** The equations will have: same slope same y-intercepts y = 2x 3 y = 2x 3 ** They are the same line! Guided Practice Without graphing, what will each system below look like? (intersecting lines, parallel lines, or same line) How many solutions will there be? (one solution, no solution, or infinitely many solutions) 1) y = x + 2 2) y = 4x + 7 3) 2x + 2y = 8 y = x + 1 y = 4x 1 y = x 4 intersecting lines parallel lines same line (lines coincide) one solution no solution infinitely many solutions You Try 1) y = x + 5 2) 3x + 2y = 5 4x + 4y = 12 3x + 2y = 6 parallel lines parallel lines no solution no solution 8 P a g e

9 SECTION 2: Solving Systems using the Substitution Method NOTE Step 1 is not necessary because the first equation is already solved for y. Guided Practice 1) 2x + 7y = 3 2) 6x 2y = 4 3) x = 1 4y y = 3x + 2 x + 5y = 3 3x 2y = 8 two lines that intersect at solves to be -4=-4 two lines that intersect at (5,-1) infinitely many solutions; (2,-1) lines will coincide You Try 1) y = 3x 2) 2x y = 8 3) x + 2y = 21 y = 2x 3 x y = 8 x + 3y = 12 two lines that intersect at solves to be 3=8 two lines that intersect at (-3,-9) no solution; parallel lines (9,1) 4) A total of 75 brownies (b) and cakes (c) were donated for a bake sale to raise money for the football team. There were four times as many brownies donated as cakes. Part A) Write a system of equations to represent this situation. b + c = 75 b = 4c Part B) Solve to determine how many of each there were. 15 cakes (c); 60 brownies (b) 9 P a g e

10 Independent Practice Name Date Period: One solution Infinite many solutions No solution Infinite many solutions No solution One solution same line; infinite many solutions intersecting lines; one solution parallel lines; no solution parallel lines; no solution Same line; Infinite many solutions Intersecting lines; One solution 10 P a g e

11 Independent Practice Name Date Period: Solve using the substitution method. (-3,5) (3,6) (0,2) (3,1) (1,-8) no solution (5, 5/3) Infinitely many solutions (4/3, 2/3) (8,2) (-1,5) (-2,-4) 11 P a g e

12 (-6,1) (3,-4) ( 5,-3) (11,9) ( ½,2) (4,2) 19. Ms. Corley wants to take her class on a trip to either the nature center or the zoo. The nature center charges $4 per student plus $95 for a 1-hour naturalist program. The zoo charges $9 per student plus $75 for a 1-hour guided tour. a) Write a sysem of equations to represent this situation. y = 4x+95 y = 9x+75 b) At what number of students will the total nature center cost be the same as the zoo s? Solve your system using the substitution method. 9x+75 = 4x+95 x = 4 students c) Ms. Corley has 22 students in her class. Determine which would cost less, the nature center or the zoo. nature center: 4(22) + 95 = $183 zoo: 9(22) + 75 = $273 Answer nature center cost less if there are 22 students going on the trip. 12 P a g e