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Coordinate Systems and Graphs Read pages 1 – 6 Pay attention to Examples: 2, 4, 6, 7, and 8 Chapter 1 Section 1

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General Form of an Equation Example 1:14x – 7y = 21 Example 2:– 3x +(1/2)y = – 6 Example 3:3x = 5 Notice that the variables are on one side of the equal sign while the constants are on the other side of the equal sign.

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Standard Form of an Equation: (Solve for y) Example 1 General Form: 14x – 7y = 21 – 7y = – 14x + 21 Standard Form: y = 2x – 3

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Standard Form of an Equation: Example 2 General Form: – 3x +(1/2)y = – 6 (1/2)y = 3x – 6 Standard Form: y = 6x – 12

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Standard Form of an Equation Question: What happens when there is no y variable? Answer: Solve for x. Example 3 3x = 5 x = 5/3

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General Linear Equation y = m x + b Where: (1) m is the slope of the line (2) b is the y-value of the y-intercept Note: The b is NOT technically the y-intercept!!!!

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How to graph the line (Blue-grey box on page 5) Plot two points and draw a line through them 1.Plot the y-intercept: ( 0, b ) 2.Plot the x-intercept. 1.Let y = 0 and solve for x 2.( x, 0 ) will be the x-intercept 3.Note: If ( 0, 0 ) is a point, then let x = 1 and solve for y and plot ( 1, y ) and draw line through the two point

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Exercise 40 (page 8) Problem Let x represent the number of years after 1969 Let y represent the amount of Rainforest (in 1,000 square miles) Amount of rainforest left given by the linear equation: y = – (25/8) x +130

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Exercise 40 part (a) Graph of the linear equation: y = – (25/8) x +130 Solution: y-intercept: ( 0, 130 ) x-intercept: ( 208/5, 0 ) = ( 41.6, 0 ) Proof of x-intercept: y = – (25/ 8)x + 130 (0) = – (25/ 8)x + 130 – 130 = – (25/ 8)x x = (– 8/ 25) (– 130 ) x = 1040/25 = 208/5 = 41.6

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Exercise 40 part (a) continued Part a: Graph of the linear Equation y = – (25/8) x +130 (0,130) (41.6, 0) x-axis y-axis

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Exercise 40 part (b) Interpret the y-intercept: ( 0, 130 ) Solution: Note that x = 0 and y = 130, and x = 0 represents 0 years after 1969 (i.e. 1969). y = 130 represents 130 thousand square miles (i.e. 130,000 square miles) The interpretation of the y-intercept is: (Alternate: In 1969, there were 130 thousand square miles of rainforest) In 1969, there were 130,000 square miles of rainforest

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Exercise 40 part (c) When were there 80,000 square miles of tropical rain forest? Solution: 80,000 square miles y = 80 y = – (25/8)x + 130 (80) = – (25/8)x + 130 – 50 = – (25/8)x x = (– 50)(– 8/25) = 16 Thus x = 16 16 years after 1969 1985

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Answer for Exercise 40 part (c) In 1985, there will be 80,000 square miles of tropical rain forest left.

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Exercise 40 part (d) How large will the forest be in 2007? Solution: 2007 – 1969 = 38 x = 38 y = – (25/8) x + 130 y = – (25/8)(38) + 130 y = 11.25 y = 11.25 11.25 thousand square miles In 2007, there were 11,250 square miles of rainforest

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Exercise 40 continuation Interpret the x-intercept: ( 41.6, 0 ) Solution: Note that x = 41.6 and y = 0. y = 0 The is no rainforest left x = 41.6 41.6 years after 1969 (i.e. 1969 + 41.6 = 2010.6) According to the model, sometime in 2010, the rainforest will disappear.

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