Chapter 1 Review Notes
Chapter 1 Linear Equations and Straight Lines Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 1 of 71 Outline 1.1 Coordinate Systems and Graphs 1.2 Linear Inequalities 1.3 The Intersection Point of a Pair of Lines 1.4 The Slope of a Straight Line Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 2 of 71 Coordinate Plane Construct a Cartesian coordinate system on a plane by drawing two coordinate lines, called the coordinate axes, perpendicular at the origin. The horizontal line is called the x-axis, and the vertical line is the y-axis. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel y y-axis O Origin Copyright © 2010 Pearson Education, Inc. x x-axis 3 of 71 Coordinate Plane: Points Each point of the plane is identified by a pair of numbers (a,b). The first number tells the number of units from the point to the y-axis. The second tells the number of units from the point to the xaxis. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 4 of 71 Example Coordinate Plane Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3). (-1,3) -1 y 3 2 (2,1) 1 x -1 (-1,-2) -2 -3 (0,-3) Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 5 of 71 Example Graph of an Equation Sketch the graph of the equation y = 2x - 1. y (2,3) x y -2 2(-2) - 1 = -5 -1 2(-1) - 1 = -3 0 2(0) - 1 = -1 1 2(1) - 1 = 1 2 2(2) - 1 = 3 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel (1,1) x (0,-1) (-1,-3) (-2,-5) Copyright © 2010 Pearson Education, Inc. 6 of 71 Linear Equation An equation that can be put in the form cx + dy = e (c, d, e constants) is called a linear equation in x and y. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 7 of 71 Standard Form of Linear Equation The standard form of a linear equation is y = mx + b (m, b constants) if y can be solved for, or x=a (a constant) if y does not appear in the equation. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 8 of 71 Example Standard Form Find the standard form of 8x - 4y = 4 and 2x = 6. (a) 8x - 4y = 4 (b) 2x = 6 8x - 4y = 4 - 4y = - 8x + 4 y = 2x - 1 2x = 6 x=3 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 9 of 71 Intercepts x-intercept: a point on the graph that has a ycoordinate of 0. This corresponds to a point where the graph intersects the x-axis. y-intercept: the point on the graph that has a xcoordinate of 0. This corresponds to the point where the graph intersects the y-axis. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 10 of 71 Graph of y = mx + b To graph the equation y = mx + b: 1. Plot the y-intercept (0,b). 2. Plot some other point. [The most convenient choice is often the x-intercept.] 3. Draw a line through the two points. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 11 of 71 Example Graph of Linear Equation Use the intercepts to graph y = 2x - 1. x-intercept: Let y = 0 y 0 = 2x - 1 (1/2,0) x = 1/2 x y-intercept: Let x = 0 y = 2(0) - 1 = -1 (0,-1) y = 2x - 1 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 12 of 71 1.2 Linear Inequalities 1. 2. 3. 4. 5. 6. 7. Definitions of Inequality Signs Inequality Property 1 Inequality Property 2 Standard Form of Inequality Graph of x > a or x < a Graph of y > mx + b or y < mx + b Graphing System of Linear Inequalities Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 13 of 71 Graph of x > a or x < a The graph of the inequality x > a consists of all points to the right of and on the vertical line x = a; x < a consists of all points to the left of and on the vertical line x = a. We will display the graph by crossing out the portion of the plane not a part of the solution. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 14 of 71 Example Graph of x > a Graph the solution to 4x > -12. First write the equation in standard form. y 4x > -12 x = -3 x > -3 x Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 15 of 71 Graph of y > mx + b or y < mx + b To graph the inequality, y > mx + b or y < mx + b: 1. Draw the graph of y = mx + b. 2. Throw away, that is, “cross out,” the portion of the plane not satisfying the inequality. The graph of y > mx + b consists of all points above or on the line. The graph of y < mx + b consists of all points below or on the line. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 16 of 71 Example Graph of y > mx + b Graph the inequality 4x - 2y > 12. First write the equation in standard form. 4x - 2y > 12 y - 2y > - 4x + 12 y < 2x - 6 x y = 2x - 6 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 17 of 71 Example Graph of System of Inequalities Graph the system of inequalities 2 x 3 y 15 4 x 2 y 12 y 0. The system in standard form is y y 2 x 5 3 2x 6 y 0. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 18 of 71 1.3 The Intersection Point of a Pair of Lines 1. 2. 3. 4. Solve y = mx + b and y = nx + c Solve y = mx + b and x = a Supply Curve Demand Curve Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 19 of 71 Solve y = mx + b and y = nx + c To determine the coordinates of the point of intersection of two lines y = mx + b and y = nx + c 1. Set y = mx + b = nx + c and solve for x. This is the x-coordinate of the point. 2. Substitute the value obtained for x into either equation and solve for y. This is the y-coordinate of the point. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 20 of 71 Example Solve y = mx + b & y = nx + c Solve the system 2 x 3 y 7 4 x 2 y 9. Write the system in standard form, set equal and solve. 2 7 y 3 x 3 y 2x 9 2 2 7 9 y x 2x 3 3 2 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel 8 41 x 3 6 41 x 16 41 9 5 y 2 16 2 8 Copyright © 2010 Pearson Education, Inc. 21 of 71 Example Point of Intersection Graph Point of Intersection: (41/16, 5/8) y y = 2x - 9/2 (41/16,5/8) x y = (-2/3)x + 7/3 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 22 of 71 1.4 The Slope of a Straight Line 1. 2. 3. 4. 5. 6. Slope of y = mx + b Geometric Definition of Slope Steepness Property Point-Slope Formula Perpendicular Property Parallel Property Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 23 of 71 Slope of y = mx + b For the line given by the equation y = mx + b, the number m is called the slope of the line. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 24 of 71 Example Slope of y = mx + b Find the slope. y = 6x - 9 m=6 y = -x + 4 m = -1 y=2 m=0 y=x m=1 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 25 of 71 Geometric Definition of Slope Geometric Definition of Slope Let L be a line passing through the points (x1,y1) and (x2,y2) where x1 ≠ x2. Then the slope of L is given by the formula y2 y1 m . x2 x1 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 26 of 71 Example Geometric Definition of Slope Use the geometric definition of slope to find the slope of y = 6x - 9. Let x = 0. Then y = 6(0) - 9 = -9. (x1,y1) = (0,-9) Let x = 2. Then y = 6(2) - 9 = 3. (x2,y2) = (2,3) 3 9 12 m 6 20 2 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 27 of 71 Steepness Property Steepness Property Let the line L have slope m. If we start at any point on the line and move 1 unit to the right, then we must move m units vertically in order to return to the line. (Of course, if m is positive, then we move up; and if m is negative, we move down.) Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 28 of 71 Example Steepness Property Use the steepness property to graph y = -4x + 3. The slope is m = -4. A point on the line is (0,3). If you move to the right 1 unit to x = 1, y must move down 4 units to y = 3 - 4 = -1. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel y (0,3) x (1,-1) y = -4x + 3 Copyright © 2010 Pearson Education, Inc. 29 of 71 Point-Slope Formula Point-Slope Formula The equation of the straight line through the point (x1,y1) and having slope m is given by y - y1 = m(x - x1). Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 30 of 71 Example Point-Slope Formula Find the equation of the line through the point 3 (-1,4) with a slope of . 5 Use the point-slope formula. 3 y 4 x 1 5 3 3 y4 x 5 5 3 17 y x 5 5 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 31 of 71 Perpendicular Property Perpendicular Property When two lines are perpendicular, their slopes are negative reciprocals of one another. That is, if two lines with slopes m and n are perpendicular to one another, then m = -1/n. Conversely, if two lines have slopes that are negative reciprocals of one another, they are perpendicular. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 32 of 71 Example Perpendicular Property Find the equation of the line through the point (3,-5) that is perpendicular to the line whose equation is 2x + 4y = 7. The slope of the given line is -1/2. The slope of the desired line is -(-2/1) = 2. Therefore, y -(-5) = 2(x - 3) or y = 2x – 11. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 33 of 71 Parallel Property Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 34 of 71 Example Parallel Property Find the equation of the line through the point (3,-5) that is parallel to the line whose equation is 2x + 4y = 7. The slope of the given line is -1/2. The slope of the desired line is -1/2. Therefore, y -(-5) = (-1/2)(x - 3) or y = (-1/2)x - 7/2. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 35 of 71 Graph of Perpendicular & Parallel Lines 2x + 4y = 7 y = 2x - 11 y = (-1/2)x - 7/2 Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc. 36 of 71
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