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### Understanding and Graphing Linear Equations: Key Concepts and Examples ###

In this chapter, you will learn to graph linear equations using both the x- and y-intercepts and the slope-intercept method. Understanding these concepts is crucial as they can illustrate real-world relationships, such as phone costs. The examples provided will detail how to find intercepts and slopes of given equations, and how to accurately graph them. Additionally, you will learn to check your work by validating points on the graph, ensuring a comprehensive understanding of linear relationships and their graphical representations. ###

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### Understanding and Graphing Linear Equations: Key Concepts and Examples ###

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  1. Chapter 7 Section 5 Graphing Linear Equations

  2. What You’ll Learn You’ll learn to graph linear equations by using the x- and y- intercepts or the slope and y intercept.

  3. Why It’s Important Rates Linear graphs are helpful in showing phone costs.

  4. Example 1 Determine the x-intercept and y-intercept of the graph of each equation. Then graph the equation. 5y – x = 10 To find the x-intercept, let y = 0. 5(0) – x = 10 - x = 10 -1 -1 x = -10 (-10,0) To find the y-intercept, let x = 0 5y – 0 = 10 5y = 10 5 5 y = 2 (0,2)

  5. Example 1: Continued • The x-intercept is -10, and the y-intercept is 2. This means that the graph intersects the x- axis at (-10, 0) and the y-axis at (0, 2). • Graph these ordered pairs. • Then draw the line that passes through these points.

  6. (0, 2) Example 1: Continued Y 8 7 6 5 4 3 2 1 0 • Graph: 5y – x = 10 (-10, 0) x -15 -14 -13-12 -11 -10 9 -8 -7 -6 -5 -4 -3 -2 -1

  7. Example 2 Determine the x-intercept and y-intercept of the graph of each equation. Then graph the equation. 2x – 4y = 8 To find the x-intercept, let y = 0. 2x – 4(0) = 8 2x = 8 2 2 x = 4 (4, 0) To find the y-intercept, let x = 0 2(0) – 4y = 8 -4y = 8 -4 -4 y = -2 (0,-2)

  8. Example 2: Continued • The x-intercept is 4, and the y-intercept is -2. This means that the graph intersects the x- axis at (4, 0) and the y-axis at (0, -2). • Graph these ordered pairs. • Then draw the line that passes through these points.

  9. (4, 0) Example 2: Continued 2 1 0 -1 -2 -3 -4 -5 • Graph: 2x – 4y = 8 x 1 2 3 4 5 6 7 8 (0, -2) Y

  10. How to Check Your Work • Look at the graph. • Choose some other point on the line and determine whether it is a solution of 2x – 4y = 8. • Try (2, -1) 2x – 4y = 8 2(2) – 4(-1) = 8 4 + 4 = 8 8 = 8

  11. Your Turn Determine the x-intercept and y-intercept of the graph of each equation. Then graph the equation. x + y = 2

  12. x + y = 2 (0, 2) (2, 0)

  13. Your Turn Determine the x-intercept and y-intercept of the graph of each equation. Then graph the equation. 3x + y = 3

  14. 3x + y = 3 (0, 3) (1, 0)

  15. Your Turn Determine the x-intercept and y-intercept of the graph of each equation. Then graph the equation. 4x – 5y = 20

  16. 4x – 5y = 20 (5, 0) (0, -4)

  17. Example 3 To mail letter in 2000, it cost $0.33 for the first ounce and $0.22 for each additional ounce. This can be represented by y = 0.33 + 0.22x. Determine the slope and y-intercept of the graph of the equation. y = mx + b y = 0.22x + 0.33 The slope is 0.22, and the y-intercept is 0.33. So the slope represents the cost per ounce after the first ounce, and the y-intercept represents the cost of the first ounce of mail.

  18. Example 4 Determine the slope and y-intercept of the graph of 10 + 5y = 2x. Write the equation in slope-intercept form to find the slope and y-intercept 10 + 5y = 2x 10 + 5y = 2x -10 = -10 5y = 2x – 10 5 5 y = ⅖x – 2 The slope is ⅖, and the y-intercept is -2.

  19. Your Turn Determine the slope and y-intercept. y = 5x + 9 m = 5, b = 9

  20. Your Turn Determine the slope and y-intercept. 4x + 3y = 6 m = - 4/3 , b = 2

  21. Example 5 Graph each equation by using the slope and y-intercept y = ⅔x – 5 y = mx + b y = ⅔x + (-5) The slope is ⅔, and the y-intercept is -5. Graph the point at (0, -5). Then go up 2 units and right 3 units. This will be the point at (3, -3). Then draw the line through points at (0, -5) and (3, -3).

  22. Example 5: Continued 3 (3, -3) 2 (0, -5)

  23. Example 5: Check • The graph appears to go through the point at (6, -1) . Substitute (6, -1) into y = ⅔x + (-5). y = ⅔x + (-5) -1 = ⅔(6) + (-5) -1 = 4 – 5 -1 = -1 Replace x with 6 and y with -1.

  24. Example 6 Graph each equation by using the slope and y-intercept. 3x + 2y = 6 First, write the equation in slope intercept form. 3x + 2y = 6 -3x = -3x 2y = -3x + 6 2 2 y = -3⁄₂x + 3 The slope is -3⁄₂ and the y-intercept is 3.

  25. Example 6: Continued • Graph the point (0, 3). • Then go up 3 units and left 2 units. • This will be the point at (-2, 6). • Then draw a line through (0, 3) and (-2, 6). • You can check your answer by substituting the coordinates of another point that appears to lie on the line, such as (2, 0).

  26. Example 6: Graph -2 (-2, 6) 3 (0, 3)

  27. Your Turn Graph each equation by using the slope and y-intercept. y= ½x + 3

  28. Answer (2, 4) (0, 3) y = ½x + 3

  29. Your Turn Graph each equation by using the slope and y-intercept. x + 4y = -8

  30. Answer (0, -2) (-4, -1) x + 4y = -8

  31. Something to Memorize • The graph of a horizontal line has a slope of 0 and no x-intercept. • The graph of a vertical line has an undefined slope and no y-intercept.

  32. Example 7 Graph each equation. y = 4 y = mx + b y = 0x + 4 No matter what the value of x, y = 4. So, all ordered pairs are of the form (x, 4). Some examples are (0, 4) and (-3, 4).

  33. 0 Example 7: Continued (-3, 4) (0, 4)

  34. Example 8 Graph each equation. x = -2 Slope is undefined, y- intercept: none No matter what the value of y, x = -2. So, all ordered pairs are of the form (-2, y). Some examples are (-2, -1) and (-2, 3).

  35. Example 8 (-2, 3) (-2, -1)

  36. Your Turn Graph each equation. y= -1

  37. y = -1 Answer

  38. Try This One Graph each equation. x = 3

  39. Answer x = 3

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