1.2 Graphs and Lines. Cartesian Coordinate System
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1 1.2 Graphs and Lines Cartesian Coordinate System Note that there is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs (a, b) of real numbers.
2 Graphs of Ax + By = C Recall a linear equation in one variable we saw earlier: Add one more variable (lets call it y) and get: Definition. A linear equation in two variables is an equation that can be written in the Standard form: Ax + By = C where A, B, and C are constants (A and B not both 0), and x and y are variables. Note: A and B cannot be zero simultaneously. However, C can very well be 0 regardless what values A and B have. Definition. A solution of an equation in two variables is an ordered pair of real numbers that satisfies the equation. Definition. The solution set of an equation in two variables is the set of all solutions of the equation. Definition. The graph of an equation is the graph of its solution set. 2
3 Example 1 5x 2y = 10 Theorem 1 (Graph of a Linear Equation in Two Variables) The graph of any equation of the form Ax + By = C (A and B not both 0) i.e. a linear equation in two variables, is a line. Also, the converse is true, i.e. any line in a Cartesian coordinate system is the graph of an equation of this form. 3
4 Three possible cases for A, and B being zero/non-zero: 1. A 0, B 0 2. A 0, B = 0 3. A = 0, B 0 Question: How to graph a linear equation in two variables? Answer: 4
5 The easiest two such points to find are: Definition. The y intercept is the y coordinate of the point where the graph crosses the y axis. Definition. The x intercept is the x coordinate of the point where the graph crosses the x axis. Question: How to find x and y intercepts? Answer: Terminology: If the x intercept is a and the y intercept is b, then the graph of the line passes through the points (a, 0) and (0, b). It is common practice to refer to both the numbers a and b, and the points (a, 0) and (0, b) as the x and y intercepts of the line. 5
6 Example 2 Graph 4x 3y = 12 6
7 Slope of a Line Definition (Slope of a Line) If a line passes through two distinct points, P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) then its slope is given by the formula m = y 2 y 1 x 2 x 1 = vertical change (rise) horizontal change (run), x 1 x 2 Example 3 Sketch a line through each pair of points, and find the slope of each line. (a) ( 1, 2), (2, 4) 7
8 (b) ( 1, 2), (3, 0) (c) (0, 3), (3, 1) (d) ( 3, 2), (1, 2) (e) ( 1, 3), ( 1, 3) 8
9 Geometric Interpretation of Slope Equations of Lines: Special Forms Slope-Intercept Form Consider a linear equation of the following form y = mx + b. 9
10 Definition (Slope-Intercept Form) The equation y = mx + b, m = slope, b = y intercept is called the slope-intercept form of an equation of a line. Example 4 (a) Find the slope and y intercept, and graph y = 3 2 x + 6 (b) Write the equation of the line with slope 2 and y intercept 4 10
11 Point-Slope Form Consider a line that passes through a fixed point (x 1, y 1 ) and has a slope m. Definition (Point-Slope Form) An equation of a line with slope m that passes through (x 1, y 1 ) is y y 1 = m(x x 1 ) which is called the point-slope form of an equation of a line Example 5 (a) Find an equation for the line that has slope 2 3 and passes through (6, 2). Write the resulting equation in the form Ax + By = C, A > 0. 11
12 (a) Find an equation for the line that passes through (2, 3) and 14, 32. Write the resulting equation in the form y = mx + b. 12
13 To summarize: Equations of a Line Applications Example 6 At a price of $12.59 per box of grapefruit, the supply is 595,000 boxes and the demand is 650,000 boxes. At a price of $13.19 per box, the supply is 695,000 boxes and the demand is 590,000 boxes. Assume that the relationship between price and supply is linear and that the relationship between price and demand is linear. (a) Find a price supply equation of the form p = mx + b. (b) Find a price demand equation of the form p = mx + b. (c) Find the equilibrium point. 13
14 14
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