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

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.

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

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

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

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!!!!

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

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

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 (0) = – (25/ 8)x – 130 = – (25/ 8)x x = (– 8/ 25) (– 130 ) x = 1040/25 = 208/5 = 41.6

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

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

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 (80) = – (25/8)x – 50 = – (25/8)x x = (– 50)(– 8/25) = 16 Thus x = years after

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

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

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 = years after 1969 (i.e = ) According to the model, sometime in 2010, the rainforest will disappear.