Chapter 1 Linear Equations and Straight Lines 2 of 71
Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5 The Method of Least Squares 3 of 71
Section 1.1 Coordinate Systems and Graphs 4 of 71
Coordinate Line Construct a Cartesian coordinate system on a line by choosing an arbitrary point, O (the origin), on the line and a unit of distance along the line. Then assign to each point on the line a number that reflects its directed distance from the origin. Slide 5 5 of 71
Example Coordinate Line Graph the points -3/5, 1/2 and 15/8 on a coordinate line. -3/5 1/2 15/8-4 -3-2 -1 0 1 2 3 4 Origin Unit length Negative numbers Positive numbers Slide 6 6 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. Origin y O y-axis x-axis x Slide 7 7 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 x-axis. The second tells the number of units from the point to the y-axis. Slide 8 8 of 71
Example Coordinate Plane Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3). (-1,3) -1 y -1 (-1,-2) 3-2 -3 2 (2,1) 1 x (0,-3) Slide 9 9 of 71
Graph of an Equation The collection of points (x,y) that satisfies an equation is called the graph of that equation. Every point on the graph will satisfy the equation if the first coordinate is substituted for every occurrence of x and the second coordinate is substituted for every occurrence of y in the equation. Slide 10 10 of 71
Example Graph of an Equation Sketch the graph of the equation y = 2x - 1. x y = 2x - 1 (x,y) -2 2(-2) - 1 = -5 (-2,-5) -1 2(-1) - 1 = -3 (-1,-3) 0 2(0) - 1 = -1 (0,-1) 1 2(1) - 1 = 1 (1,1) 2 2(2) - 1 = 3 (2,3) (1,1) (0,-1) (-1,-3) (-2,-5) y (2,3) x Slide 11 11 of 71
General Linear Equation An equation that can be written in the form cx + dy = e (c, d, e constants) is called a linear equation in x and y. Slide 12 12 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. Slide 13 13 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 Slide 14 14 of 71
Graph of x = a The equation x = a graphs into a vertical line a units from the y-axis. y x = 2 x = -3 y x x Slide 15 15 of 71
Intercepts x-intercept: the point where the graph intersects the x-axis. This corresponds to a point on the graph that has a y-coordinate of 0. Similarly y-intercept: the point where the graph intersects the y-axis. This corresponds to a point on the graph that has a x-coordinate of 0. Slide 16 16 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. Slide 17 17 of 71
Example Graph of Linear Equation Use the intercepts to graph y = 2x - 1. x-intercept: Let y = 0 y 0 = 2x - 1 x = 1/2 y-intercept: Let x = 0 y = 2(0) - 1 = -1 y = 2x - 1 (0,-1) (1/2,0) x Slide 18 18 of 71
Summary Section 1.1 Ø Cartesian coordinate systems associate a number with each point of a line and associate a pair of numbers with each point of a plane. Ø The collection of points in the plane that satisfy the equation ax + by = c lies on a straight line. Ø After this equation is put into one of the standard forms y = mx + b or x = a, the graph is easily drawn. Slide 19 19 of 71
Section 1.4 The Slope of a Straight Line 20 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. Slide 21 21 of 71
Example Slope of y = mx + b Find the slope. y = 6x - 9 y = -x + 4 y = 2 y = x m = 6 m = -1 m = 0 m = 1 Slide 22 22 of 71
Geometric Definition of Slope Geometric Definition of Slope Let L be a line passing through the points (x 1,y 1 ) and (x 2,y 2 ) where x 1 x 2. Then the slope of L is given by the formula y2 y1 m =. x x 2 1 Slide 23 23 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. (x 1,y 1 ) = (0,-9) Let x = 2. Then y = 6(2) - 9 = 3. (x 2,y 2 ) = (2,3) m 3 ( 9) 12 = = = 2 0 2 6 Slide 24 24 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.) Slide 25 25 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 y (0,3) (1,-1) x down 4 units to y = 3-4 = -1. y = -4x + 3 Slide 26 26 of 71
Point-Slope Formula Point-Slope Formula The equation of the straight line through the point (x 1,y 1 ) and having slope m is given by y - y 1 = m(x - x 1 ). Slide 27 27 of 71
Example Point-Slope Formula Find the equation of the line that passes through (-1,4) with a slope of 3. Use the point-slope formula. 3 y 4= 1 5 3 3 y 4 = x 5 5 3 17 y = x+ 5 5 5 ( x ( )) Slide 28 28 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. Slide 29 29 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. Slide 30 30 of 71
Parallel Property Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel. Slide 31 31 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. Slide 32 32 of 71
Graph of Perpendicular & Parallel Lines 2x + 4y = 7 y = 2x - 11 y = (-1/2)x - 7/2 Slide 33 33 of 71
Summary Section 1.4 - Part 1 Ø The slope of the line y = mx + b is the number m. It is also the ratio of the difference between the y-coordinates and the difference between the x-coordinates of any pair of points on the line. Ø The steepness property states that if we start at any point on a line of slope m and move 1 unit to the right, then we must move m units vertically to return to the line. Slide 34 34 of 71
Summary Section 1.4 - Part 2 Ø The point-slope formula states that the line of slope m passing through the point (x 1, y 1 ) has the equation y - y 1 = m(x - x 1 ). Ø Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is 1. Slide 35 35 of 71
Section 1.3 The Intersection Point of a Pair of Lines 36 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. Slide 37 37 of 71
Example Solve y = mx + b & y = nx + c Solve the system 2x + 3y = 7 4x _ 2y = 9. Write the system in standard form, set equal and solve. - 2 7 y = x + 3 3 9 y = 2x _ 2 2 7 9 y = x+ = 2x- 3 3 2 8 41 x = 3 6 41 x = 16 41 9 5 y = 2. _ = 16 2 8 Slide 38 38 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 Slide 39 39 of 71
Solve y = mx + b and x = a To determine the coordinates of the point of intersection of two lines: y = mx + b and x = a 1. The x-coordinate of the point is x = a. 2. Substitute x = a into y = mx + b and solve for y. This is the y-coordinate of the point. Slide 40 40 of 71
Example Solve y = mx + b & x = a Find the point of intersection of the lines y = 2x - 1 and x = 2. The x-coordinate of the point is x = 2. Substitute x = 2 into y = 2x - 1 to get the y-coordinate. y = 2(2) - 1 = 3 Intersection Point: (2,3) y = 2x - 1 y (2,3) x = 2 x Slide 41 41 of 71
Summary Section 1.3 Ø The point of intersection of a pair of lines can be obtained by first converting the equations to standard form and then either equating the two expressions for y or substituting the value of x from the form x = a into the other equation. Slide 42 42 of 71
Section 1.2 Linear Inequalities 43 of 71
Definitions of Inequality Signs Ø a < b means a lies to the left of b on the number line (if the number line is the x-axis) or a lies below b on the number line (if the number line is the y-axis). Ø a < b means a = b or a < b. Ø Similarly, a > b means a lies to the right of b or above b on the number line (depending on the axis). Ø a > b means a = b or a > b. Slide 44 44 of 71
Inequality Signs Example -4-3 -2-1 0 1 2 3 4 Which of the following statements are true? 1 < 4 True -1 > -4 True 2 < 3 True 0 < -2 False 3 > 3 True Slide 45 45 of 71
Inequality Property 1 Inequality Property 1 Suppose that a < b and that c is any number. Then a + c < b + c. In other words, the same number can be added or subtracted from both sides of the inequality. Note: Inequality Property 1 also holds if < is replaced by >, < or >. Slide 46 46 of 71
Example Inequality Property 1 Solve (?) the inequality x + 5 < 2. Subtract 5 from both sides to isolate the x on the left. x + 5 < 2 x + 5-5 < 2-5 x < -3 The values of x for which the inequality holds are exactly those x less than or equal to 3. Slide 47 47 of 71
Inequality Property 2 Inequality Property 2 2A. If a < b and c is positive, then ac < bc. 2B. If a < b and c is negative, then ac > bc. Note: Inequality Property 2 also holds if < is replaced by >, < or >. Slide 48 48 of 71
Example Inequality Property 2 Solve the inequality -3x + 1 > 7. Subtract 1 from both sides to isolate the x term on the left. -3x + 1 > 7-3x + 1-1 > 7-1 -3x > 6 Divide by -3, or multiply by -1/3 to isolate the x. x < -2 Slide 49 49 of 71
Standard Form of Linear Inequality A linear inequality of the form cx + dy < e can be written in the standard form 1. y < mx + b or y > mx + b if d 0, or 2. x < a or x > a if d = 0. Note: The inequality signs can be replaced by >, < or >. Slide 50 50 of 71
Example Linear Inequality Standard Form Find the standard form of 5x - 3y < 6 and 4x > -8. (a) 5x - 3y < 6 (b) 4x > -8 5x - 3y < 6-3y < - 5x + 6 y > (5/3)x - 2 4x > -8 x > -2 Slide 51 51 of 71
Graph of x > a or x < a The graph of the inequality Ø x > a consists of all points to the right of and on the vertical line x = a; Ø x < a consists of all points to the left of and on the vertical line x = a. Ø We will display the graph by crossing out the portion of the plane not a part of the solution. Slide 52 52 of 71
Example Graph of x > a Graph the solution to 4x > -12. First write the equation in standard form. 4x > -12 x > -3 y x = -3 x Slide 53 53 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. 3. 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. Slide 54 54 of 71
Example Graph of y > mx + b Graph the inequality 4x - 2y > 12. First write the equation in standard form. 4x - 2y > 12-2y > - 4x + 12 y < 2x - 6 y x y = 2x - 6 Slide 55 55 of 71
Example Graph of System of Inequalities Graph the system of inequalities The system in standard form is y y 2 x 3 + 5 2x 6 y 0. y < -2/3 x + 5 2x + 3y 15 _ 4x 2y 12 y 0. y > 2 x - 6 Feasible set y < 0 Slide 56 56 of 71
Summary Section 1.2 - Part 1 Ø The direction of the inequality sign in an inequality is unchanged when a number is added to or subtracted from both sides of the inequality, or when both sides of the inequality are multiplied by the same positive number. Ø The direction of the inequality sign is reversed when both sides of the inequality are multiplied by the same negative number. Slide 57 57 of 71
Summary Section 1.2 - Part 2 Ø The collection of points in the plane that satisfy the linear inequality ax + by < c or ax + by > c consists of all points on and to one side of the graph of the corresponding linear equation. Ø After this inequality is put into standard form, the graph can be easily pictured by crossing out the half-plane consisting of the points that do not satisfy the inequality. Slide 58 58 of 71
Summary Section 1.2 - Part 3 Ø The feasible set of a system of linear inequalities (that is, the collection of points that satisfy all the inequalities) is best obtained by crossing out the points not satisfied by each inequality. The feasible set associated to the system of the previous example is a three-sided unbounded region. Slide 59 59 of 71