7.1 – Cartesian Co-ordinate System & Linear Equations in Two Variables x-axis; y-axis; origin; quadrants Co-ordinates; ordered pair Linear Equations in Two variables Standard form ax+by = c Solutions as ordered pairs; number of solutions Graph of equation; collinear points Calculator – window, scale

Cartesian Co-ordinate System Terminology x-co-ordinate ( x , y ) is an ordered pair y-co-ordinate Cartesian Co-ordinate System Terminology y-axis II I 2nd quadrant 1st quadrant ( - , + ) ( + , + ) (x , y) x-axis III IV 3rd quadrant 4th quadrant ( - , - ) ( + , - ) origin ( 0 , 0 ) The ordered pair ( x , y ) corresponds to a point in the cartesian plane.

Cartesian Co-ordinate System Examples x-co-ordinate ( x , y ) is an ordered pair y-co-ordinate Cartesian Co-ordinate System Examples y-axis (-,+) II I (+,+) 2 (-1,2) (1,2) 1 (-2,1) (2,1) -2 -1 0 1 2 x-axis (-2,-1) (2,-1) -1 (-1,-2) (1,-2) -2 (-,-) III IV (+,-) Plot / graph the ordered pairs (1,2); (-1,2) (-2,1); (2,1) (2,-1); (-2,-1) (-1,-2); (1,-2) The ordered pair ( x , y ) corresponds to a point in the cartesian plane.

Linear Equations in Two Variables x and y Examples 3x – 4y +3 = 0 – 4y+ 3x = -3 3x = 4y – 3 4y – 3 – 3x = 0 3 + 3x = 4y 3 + x + 2x = 4y Standard form 3x – 4y = -3 or -3x + 4y = 3 Standard form: ax + by = c where a, b, and c are real numbers. Example x = 4y + 5 x = 4y x – 5 = 0 y = 7 Standard Form x – 4y = 5 x – 4y = 0 c = 0 x = 5 b = 0 y = 7 a = 0

Linear equations in x & y – solutions as ordered pairs Consider 2x + y = 4 If x = 1 find y 2(1) + y = 4 and y = 2 Or if y = 2 find x 2x + (2) = 4 and x = 1 The pair of #’s: x = 1 & y = 2 satisfy 2x + y = 4 y 3 2 1 -3 -2 -1 0 1 2 3 x -1 -2 -3 Write as an ordered pair (1 , 2) Say (1 , 2) is a solution to 2x + y = 4

Linear equations in x & y – solutions as ordered pairs Consider 2x + y = 4 again If x = 2 find y 2(2) + y = 4 and y = 0 Or if y = 0 find x 2x + (0) = 4 and x = 2 The pair of #’s: x = 2 & y = 0 satisfy 2x + y = 4 y 3 2 1 -3 -2 -1 0 1 2 3 x -1 -2 -3 Previous solution (1,2) Write as an ordered pair (2 , 0) Say (2 , 0) is a solution to 2x + y = 4 Another solution(2,0)

Linear equations in x & y – solutions as ordered pairs Consider 2x + y = 4 and again If x = 1.5 find y 2(1.5) + y = 4 and y = 1 The pair of #’s: x = 1.5 & y = 1 satisfy 2x + y = 4 y 3 2 1 -3 -2 -1 0 1 2 3 x -1 -2 -3 (1.5 , 1) is a solution to 2x + y = 4 The points (1,2); (1.5,1) and (2,0) are collinear – they are in a straight line.

Linear equations in x & y – solutions as ordered pairs Consider 2x + y = 4 yet again Other solutions ? y = -1 and x = y = -2 and x = x = 0 and y = y 3 2 1 -3 -2 -1 0 1 2 3 x -1 -2 -3 2.5 (2.5,-1) 3 (3, -2) 4 (0,4) (-1, 2) is not collinear with other points. Is (-1,2) a solution to 2x + y = 4? 2(-1) + 2 = 4 is false, so answer is no

Linear equations in x & y – solutions as ordered pairs Consider 2x + y = 4 once last time How many solutions does the equation 2x + y = 4 have? Infinitely many. Every point on the graphed line are solutions to the equation. y 3 2 1 -3 -2 -1 0 1 2 3 x -1 -2 -3

Definition of a graph A graph of an equation is a set of points whose co-ordinates satisfy the equation. y 3 2 1 -3 -2 -1 0 1 2 3 x -1 -2 -3 Example: y = 1 + x Standard form: x – y = 1 The graph of x – y = 1 has collinear points – all points lie on the straight line. Non-collinear points are not solutions of the linear equation x – y = 1 . They lie off the line.

Calculator tips - see p447 WINDOW (top row, 2nd from left) For example, set the window size to Xmin = -20 [use the (-) key, not the – key] Xmax = 20 Xscl = 5 Ymin = 0 Ymax = 10 Yscl = 2 Practice setting different window sizes and scales to see how the axes and tic marks change on your calculator. GRAPH (top right) to see axes