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Chapter 4 Linear Equations and Inequalities in Two Variables

Chapter 4 Linear Equations and Inequalities in Two Variables. § 4.1. Graphing Linear Equations in Two Variables. The Rectangular Coordinate System. In the rectangular coordinate system, the horizontal number line is the x-axis .

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Chapter 4 Linear Equations and Inequalities in Two Variables

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  1. Chapter 4Linear Equations and Inequalities in Two Variables

  2. §4.1 Graphing Linear Equations in Two Variables

  3. The Rectangular Coordinate System In the rectangular coordinate system, the horizontal number line is the x-axis. The vertical number line is the y-axis. The point of intersection of these axes is their zero points, called the origin. The axes divide the plane into 4 quarters, called quadrants. y-axis 2nd quadrant 1st quadrant x-axis 3rd quadrant 4th quadrant Blitzer, Introductory Algebra, 5e – Slide #3 Section 4.1

  4. The Rectangular Coordinate System Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers (x,y). Note the word “ordered” because order matters. The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number, the y-coordinate, denotes vertical distance and direction along a line parallel to the y-axis or along the y-axis Itself. In plotting points, we move across first (either left or right), and then move either up or down, alwaysstarting from the origin. Blitzer, Introductory Algebra, 5e – Slide #4 Section 4.1

  5. Plotting Points EXAMPLE Plot the points (3,2) and (-2,-4). SOLUTION (3,2) (-2,-4) Blitzer, Introductory Algebra, 5e – Slide #5 Section 4.1

  6. The Graph of an Equation The graph of an equation in two variables is the set of points whose coordinates satisfy the equation. An ordered pair of real numbers (x,y) is said to satisfy the equation when substitution of the x and y coordinates into the equation makes it a true statement. For example, in the equation y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this point the sentence reads 8 = 8, which is true. The ordered pair (2,3) is not a solution. When we substitute this point, the sentence reads 3 = 10, which is not true. Blitzer, Introductory Algebra, 5e – Slide #6 Section 4.1

  7. Solution of an Equation in Two Variables Remember…. If when the x-coordinate of the ordered pair is substituted for x and the y-coordinate of the ordered pair is substituted for y, we obtain a true statement- then the ordered pair is a solution to the equation. Blitzer, Introductory Algebra, 5e – Slide #7 Section 4.1

  8. Solution of an Equation in Two Variables Example: Given the equation 2x + 3y = 18, determine if the ordered pair (3, 4) is a solution to the equation. We substitute 3 in for x and 4 in for y. 2(3) + 3 (4) ? 18 6 + 12 ? 18 18 = 18 True. Therefore, the ordered pair (3, 4) is a solution to the equation 2x + 3y = 18. Blitzer, Introductory Algebra, 5e – Slide #8 Section 4.1

  9. Finding Solutions of an Equation Find five solutions to the equation y = 3x + 1. Start by choosing some x values and then computing the corresponding y values. If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5) If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2) If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1) If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4) If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7) Blitzer, Introductory Algebra, 5e – Slide #9 Section 4.1

  10. Graph of the Equation Plot the five ordered pairs to obtain the graph of y = 3x + 1 (2,7) (1,4) (0,1) (-1,-2) (-2,-5) Blitzer, Introductory Algebra, 5e – Slide #10 Section 4.1

  11. Graphing an Equation EXAMPLE Graphs in the rectangular coordinate system can also be used to tell a story. Try to select the graph that best illustrates the story of the population of the U.S.A. (a) (b) (c) Population Population Population Years Years Years SOLUTION Graph (c) Blitzer, Introductory Algebra, 5e – Slide #11 Section 4.1

  12. Linear Equations Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the form ax + b = 0, where a and b are real numbers and a is not equal to 0. An example of a linear equation in x is 4x + 2 = 6. Linear equations in x are first degree equations in the variable x. Blitzer, Introductory Algebra, 5e – Slide #12 Section 4.1

  13. Comparison of Graphs of Linear Equations The graph of y = mx + b is the graph of y = mx shifted b units up when b is a positive number. The graph of y = mx + b is the graph of y = mx shifted b units down when b is a negative number. More on lines to come in this chapter.… Blitzer, Introductory Algebra, 5e – Slide #13 Section 4.1

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