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CHAPTER 1 LINEAR EQUATION AND FUNCTION 2 nd Semester, S.Y 2013 – 2014

CHAPTER 1 LINEAR EQUATION AND FUNCTION 2 nd Semester, S.Y 2013 – 2014. Rectangular Coordinates. Y-Axis. QUADRANT II (-x, y). QUADRANT I (x, y). Origin. QUADRANT III (-x, -y). QUADRANT IV (x, -y). X-Axis. 3. 2. 1. -3. -2. -1. 1. 2. 3. -1. -2. -3.

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CHAPTER 1 LINEAR EQUATION AND FUNCTION 2 nd Semester, S.Y 2013 – 2014

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  1. CHAPTER 1 LINEAR EQUATION AND FUNCTION 2nd Semester, S.Y 2013 – 2014

  2. Rectangular Coordinates Y-Axis QUADRANT II (-x, y) QUADRANT I (x, y) Origin QUADRANT III (-x, -y) QUADRANT IV (x, -y) X-Axis

  3. 3 2 1 -3 -2 -1 1 2 3 -1 -2 -3 Rectangular Coordinates Y axis (1, 2) Ordered Pair The X axis x

  4. Let’s Check Your Understanding! • What is the ordered pair located at Quadrant I? • What is the order pair plotted in Quadrant IV? • What is the point of origin? • What is x-coordinate in the ordered pair (-5, -3)? • What is the value of y or ordinate in the ordered pair (-4, 5)?

  5. What is Linear Equation? • A linear equation is an equation whose graph or solutions form a straight line on a coordinate plane. • Real life economic situations of linear equations include demand and supply analysis, cost and revenue, consumption and savings, production, stock exchange, etc.

  6. Ax + By + C = 0 A, B and C are constants. The are two variables (x and y) The variables are added or subtracted. At least one of A and B is nonzero Besides x and y, other commonly used variables are m and n, a and b, and r and s. There are no radicals in the equation. Every linear equation graphs as a straight line. Identifying a Linear Equation

  7. Examples of Linear Equations Equation is in Ax + By + C = 0form Rewrite with both variables on left side … x + 6y =3 B=0 … x + 0 y =1 Multiply both sides of the equation by -1 … 2a – b = -5 Multiply both sides of the equation by 3 … 4x –y =-21 2x + 4y =8 6y = 3 – x x = 1 -2a + b = 5

  8. Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By + C = 0: 4x2 + y = 5 xy + x = 5 s/r + r = 3 The exponent is 2 There is a radical in the equation Variables are multiplied Variables are divided

  9. The x-intercept is the point where a line crosses the x-axis. The general form of the x-intercept is (x, 0). The y-coordinate will always be zero. The y-intercept is the point where a line crosses the y-axis. The general form of the y-intercept is (0, y). The x-coordinate will always be zero. X and Y – Intercepts

  10. For the equation 2x + y = 6, we know that y must equal 0. What must x equal? Plug in 0 for y and simplify. 2x + 0 = 6 2x = 6 x = 3 So (3, 0) is the x-intercept of the line. Finding the x-intercept

  11. For the equation 2x + y = 6, we know that x must equal 0. What must y equal? Plug in 0 for x and simplify. 2(0) + y = 6 0 + y = 6 y = 6 So (0, 6) is the y-intercept of the line. Finding the y-intercept

  12. To find the x-intercept, plug in 0 for y. To find the y-intercept, plug in 0 for x. To summarize….

  13. x-intercept: Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5 (-5, 0) is the x-intercept y-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y = y (0, ) is the y-intercept Find the x and y- intercepts of

  14. Example: Graph the equation Solve for y first. - Add 5x to both sides y = 5x + 2 The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane. Graphing Equations

  15. x y Graphing Equations Graph y = 5x + 2

  16. Graphing Equations Graph • Solve for y first 4x - 3y =12 Subtract 4x from both sides -3y = -4x + 12 Divide by -3 y = x + Simplify y = x – 4 • The equation y = x - 4 is in slope-intercept form, . The y -intercept is -4 and the slope is . Graph the line on the coordinate plane.

  17. Slope is the ratio of vertical change to the horizontal change (rise/run) of a line. Slope in a linear equation shows if the line is ascending (positive) or descending (negative). Slope can also show the rate of change. The letter m is used to represent slope in a formula. Slope

  18. Slope Slope of a Straight Line Negative Undefined Slope Positive Zero Slope

  19. Let’s Check Your Understanding Study the four accompanying diagrams. Consider the following statements and indicate which diagram matches each statement. Which variable would appear on the horizontal and which on the vertical axis? In each of these statements, is the slope positive, negative, zero, or infinity? a. If the price of movies increases, fewer consumers go to see movies. b. More experienced workers typically have higher incomes than less experienced workers. c. Whatever the temperature outside, people consume the same number of hotdogs per day. d. Consumers buy more frozen yogurt when the price of ice cream goes up. e. Research finds no relationship between the number of diet books purchased and the number of pounds lost by the average dieter. f. Regardless of its price, consumers in Lingayen buy the same quantity of salt.

  20. Let’s Check Your Understanding Compute the slope of the following diagrams.

  21. Standard Form Slope-Intercept Form Two-point Form Intercept Form Point-Slope Form Vertical Line Horizontal Line Equations of a Line

  22. Standard Form Standard form for linear equations is Ax + By + C = 0 Graph the line Write the following equations in standard form. A. B. C. D.

  23. Slope intercept form is . This form makes it easy to find the slope (m) and the y-intercept (b). Working with this form is simple, so it is used more often than other forms. Example: y= ¾x + 3 *¾ is the slope. * 3 is the point where the line crosses the Y-axis. Slope Intercept Form

  24. Determine the Slope and the Y-intercept. Then draw a graph. Let’s Check Your Understanding!

  25. Which of the following equations represent demand curves, which represent supply curves? (Assume y represents price and x represents quantity) Let’s Check Your Understanding!

  26. The two-point form for a straight line, is generally most convenient for determining the equation of a straight line when two of its points are given. Two-point Form

  27. Find the equation of the line passing through the points(2, 3) and (3, 7). Find the equation of the line passing through the points (3, 4) and (-5, 2). Two-point Form

  28. Two-point Form – Econ Application • Suppose that the market demand for Android tablets per week in Metro Manila is given by the following demand schedule. Derive the demand equation.

  29. Two-point Form –Econ Application • When the price is P500, 50 MP3 players of a fixed type are available for sale; when the price is P750, 100 of the MP3 players are available. What is the supply equation?

  30. Since the slope of a nonverticalline is , the equation for point-slope form can be written Generally most convenient for determining the equation of a straight line when one of the points on a line and its slope are given. Point-Slope Form

  31. Find the equation of the line passing through the point ( -1, 2) and having slope – 4. Point-Slope Form

  32. Generally most convenient for determining the equation of a straight line when its two intercepts are given. The formula is where b is y-intercept and a is x-intercept Intercept Form

  33. Find the equation of the line having intercepts (0, -6) and (4, 0) or Intercept Form

  34. Since the slope of a vertical line is undefined, the above formulas are not appropriate for obtaining the equations of vertical lines. A vertical line passing through the point (x1, y1) has the equation Vertical Lines

  35. Find the equation of the vertical line passing through the point (5, -4). Vertical Lines

  36. Find the equation of the vertical line passing through the point (5, -4). Vertical Lines

  37. Since a horizontal line has zero slope, its equation maybe obtained from the two-point, point-slope, or slope-intercept forms. A horizontal line passing through the point (x1, y1) has the equation Horizontal Lines

  38. Find the equation of the vertical line passing through the point (2, 4). Horizontal Lines

  39. A system of linear equations consists of two or more linear equations with the same variables. Any two lines in a plane are either parallel or intersecting lines. Lines that intersect at right angles re perpendicular Parallel, Perpendicular and Intersecting Lines

  40. Parallel lines are two lines in a plane that have equal angles of inclination and therefore have equal slopes and conversely. Two lines with different -intercepts are parallel if they have the same slope. Also, two vertical lines are parallel. Coincident (equivalent equations) are two lines having the same slope and the same y-intercept. Parallel and Coincident Lines

  41. Parallel Lines

  42. Coincident Lines

  43. Let’s Check Your Understanding • Determine whether the line given by • is parallel to the line given by

  44. Perpendicular Lines • Lines that intersect at right angles • Two lines that are perpendicular have slopes which are the negative reciprocals of each other, and conversely. • Two lines are perpendicular if the product of their slopes is -1 or if one line is vertical and the other is horizontal.

  45. Perpendicular Lines • Determine whether the graphs of are perpendicular

  46. Perpendicular Lines

  47. Conditions for Two Lines (Parallel or Perpendicular • The conditions for two lines to be parallel or perpendicular can also be stated as follows: • The Linesand are parallel if and perpendicular if

  48. Conditions for Two Lines (Parallel or Perpendicular Determine whether the line given byis parallel to the line given by 2. Determine whether the line given byis parallel to the line given by

  49. Conditions for Two Lines (Parallel or Perpendicular Determine whether the line given byis perpendicular to the line given by 2. Determine whether the line given byis perpendicular to the line given by

  50. Two straight nonparallel lines that intersect in exactly one point. The point of intersection of two lines can be found by solving their equations simultaneously. Two lines which intersect corresponds to the algebraic condition that their equations are independent and consistent, and therefore have a simultaneous solution. Intersecting Lines

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