1. Linear Equations 02/11/12 lntaylor ©

2. Table of Contents Learning Objectives Graph linear equations with slope intercept method Graph linear equations using intercepts Reading a graph Rewrite linear equations from standard to slope intercept form Constructing a line from 2 points Determining if a line is on a point Graphing Inequalities 02/11/12 lntaylor ©

3. Learning Objectives TOC 02/11/12 lntaylor ©

4. Linear Equations In these sections you will learn/review how to: –Meet or exceed proficiency in AF 6.0, 7.0 and 8.0 –Pass questions regarding these standards on District or State Tests –Understand Algebra Functions based on how x values change y values –Understand Linear Equations vs. Linear Inequalities TOC 02/11/12 lntaylor ©

5. Slope Intercept Method TOC 02/11/12 lntaylor ©

6. Linear Equations Slope intercept method y = mx + b –You are given certain information in a slope intercept equation Given a slope which is called m M is a fraction that denotes Δy / Δx Δy informs whether the slope will be + or – Δx informs whether the slope is steep or flat Given a y intercept (y i or b) which informs where a line crosses the y axis TOC 02/11/12 lntaylor ©

7. 0,0 Graph y = 2x + 1 Step 1 – Identify information Identify slope m; Δ ’s; b m = 2 1 Δ y = 2; Δ x = 1 b= +1 Step 2 - Draw Locate 0,0 Go up or down b Put a dot for the 1 st point Go up or down Δ y Go right Δ x Put a dot for 2 nd point Connect dots Label the line y = 2x + 1 TOC 02/11/12 lntaylor ©

8. Now you try! y = 3x - 5 TOC 02/11/12 lntaylor ©

9. 0,0 Graph y = 3x - 5 Step 1 – Identify information Identify slope m; Δ ’s; b m = 3 1 Δ y = 3; Δ x = 1 b= - 5 Step 2 - Draw Locate 0,0 Go up or down b Put a dot for the 1 st point Go up or down Δ y Go right Δ x Put a dot for 2 nd point Connect dots Label the line y = 3x - 5 TOC 02/11/12 lntaylor ©

10. Now you try! y = - ¾ x + 2 TOC 02/11/12 lntaylor ©

11. 0,0 Graph y = - ¾ x + 2 Step 1 – Identify information Identify slope m; Δ ’s; b m = - 3 4 Δ y = - 3; Δ x = 4 b = + 2 Step 2 - Draw Locate 0,0 Go up or down b Put a dot for the 1 st point Go up or down Δ y Go right Δ x Put a dot for 2 nd point Connect dots Label the line y = - ¾ x + 2 TOC 02/11/12 lntaylor ©

12. Now you try a “hard one” ! y = - ¾ x - ½ TOC 02/11/12 lntaylor ©

13. 0,0 Graph y = - ¾ x - ½ Step 1 – Identify information Identify slope m; Δ ’s; b m = - 3 4 Δ y = - 3; Δ x = 4 b = - ½ Step 2 - Draw Locate 0,0 Go up or down b Put a dot for the 1 st point Go up or down Δ y Go right Δ x Put a dot for 2 nd point Connect dots Label the line y = - ¾x - ½ TOC 02/11/12 lntaylor ©

14. Intercepts Method TOC 02/11/12 lntaylor ©

15. Linear Equations Intercepts method using standard form ax + by = c –You are given certain information in a standard form equation Given a constant c X intercept is determined by dividing c/a or the x coefficient Y intercept is determined by dividing c/b or the y coefficient M is determined by reversing the sign for a and dividing by b or –a/b TOC 02/11/12 lntaylor ©

16. 0,0 Graph 3x + 4y = 12 Step 1 – Identify information Identify yi and xi yi = 12/4 or 3 xi = 12/3 or 4 Step 2 - Draw Locate 0,0 Go up or down yi Put a dot for the 1 st point Locate 0,0 Go right or left xi Put a dot for 2 nd point Connect dots Label the line 3x + 4y = 12 TOC 02/11/12 lntaylor ©

17. Now you try! 2x + 5y = 10 TOC 02/11/12 lntaylor ©

18. 0,0 Graph 2x + 5y = 10 Step 1 – Identify information Identify yi and xi yi = 10/5 or 2 xi = 10/2 or 5 Step 2 - Draw Locate 0,0 Go up or down yi Put a dot for the 1 st point Locate 0,0 Go right or left xi Put a dot for 2 nd point Connect dots Label the line 2x + 5y = 10 TOC 02/11/12 lntaylor ©

19. Now you try! -2x - 5y = 10 TOC 02/11/12 lntaylor ©

20. 0,0 Graph - 2x - 5y = 10 Step 1 – Identify information Identify yi and xi yi = 10/-5 or - 2 xi = 10/- 2 or - 5 Step 2 - Draw Locate 0,0 Go up or down yi Put a dot for the 1 st point Locate 0,0 Go right or left xi Put a dot for 2 nd point Connect dots Label the line - 2x - 5y = 10 TOC 02/11/12 lntaylor ©

21. Last one! 2x - 4y = 12 TOC 02/11/12 lntaylor ©

22. 0,0 Graph 2x - 4y = 12 Step 1 – Identify information Identify yi and xi yi = 12/- 4or - 3 xi = 12/2 or 6 Step 2 - Draw Locate 0,0 Go up or down yi Put a dot for the 1 st point Locate 0,0 Go right or left xi Put a dot for 2 nd point Connect dots Label the line 2x - 4y = 12 TOC 02/11/12 lntaylor ©

23. Reading graphs TOC 02/11/12 lntaylor ©

24. 0,0 Find equation Step 1 – Identify information Starting at far left of the graph find two points where diagonal, horizontal and vertical lines meet Dot the first intersection Continue to next intersection Put another dot Step 2 – Find the slope With the 1st dot go up / down and count how many grid lines you pass Continue to the right until you hot the 2 nd dot and count how many grid lines you pass You have your m Step 3 - Find b b is where the diagonal crosses the y axis + 5 4 y = x + 3 TOC 02/11/12 lntaylor ©

25. Now you try! TOC 02/11/12 lntaylor ©

26. 0,0 Find equation Step 1 – Identify information Starting at far left of the graph find two points where diagonal, horizontal and vertical lines meet Dot the first intersection Continue to next intersection Put another dot Step 2 – Find the slope With the 1st dot go up / down and count how many grid lines you pass Continue to the right until you hot the 2 nd dot and count how many grid lines you pass You have your m Step 3 - Find b b is where the diagonal crosses the y axis + 3 5 y = x - 2 TOC 02/11/12 lntaylor ©

27. Now you try a negative slope! TOC 02/11/12 lntaylor ©

28. 0,0 Find equation Step 1 – Identify information Starting at far left of the graph find two points where diagonal, horizontal and vertical lines meet Dot the first intersection Continue to next intersection Put another dot Step 2 – Find the slope With the 1st dot go up / down and count how many grid lines you pass Continue to the right until you hot the 2 nd dot and count how many grid lines you pass You have your m Step 3 - Find b b is where the diagonal crosses the y axis - 8 3 y = x - 2 ⅓ TOC 02/11/12 lntaylor ©

29. Rewriting Equations TOC 02/11/12 lntaylor ©

30. Rewriting linear equations Given equation 2x + 3y = 12 Divide everything by the y coefficient Move the slope to the other side and do not forget to change the sign! 2x + 3y = 122x2x312 3 + y = - + 4 TOC 02/11/12 lntaylor ©

31. Now you try! TOC 02/11/12 lntaylor ©

32. Rewriting linear equations Given equation - 3x + 5y = 15 Divide everything by the y coefficient Move the slope to the other side and do not forget to change the sign! Rewrite sign if necessary! - 3x + 5y = 15- 3x5155+ y = - + 3 y = 3x + 3 5 TOC 02/11/12 lntaylor ©

33. Now you try a hard one! TOC 02/11/12 lntaylor ©

34. Rewriting linear equations Given equation - 3x + 5y = - 15 Divide everything by the y coefficient Move the slope to the other side and do not forget to change the sign! Rewrite sign if necessary! - 3x + 5y = - 15- 3x5- 155+ y = - - 3 y = 3x - 3 5 TOC 02/11/12 lntaylor ©

35. Rewriting slope intercept to standard form TOC 02/11/12 lntaylor ©

36. Rewriting linear equations Given equation y = ¾ x + 7 Multiply everything by denominator Move the slope to the other side and do not forget to change the sign! Rewrite sign if necessary! y = 3x3x 4 + 7 4(4) + 7y = 3x - + 28 - 3x + 4y = 28 TOC 02/11/12 lntaylor ©

37. Now you try! TOC 02/11/12 lntaylor ©

38. Rewriting linear equations Given equation y = ⅓ x - 7 Multiply everything by denominator Move the slope to the other side and do not forget to change the sign! Rewrite sign if necessary! y = 1x1x 3 - 7 3(3) - 7y = 1x - + - 21 - x + 3y = - 21 TOC 02/11/12 lntaylor ©

39. Constructing a line given two points TOC 02/11/12 lntaylor ©

40. 0,0 Find equation (-3,2) (4,3) Step 1 – Identify information Start at 0,0; find first point Dot the first point Start at 0,0; find second point Put another dot Step 2 – Find the slope Find Δy (y 1 – y 2 ) Find Δx (x 1 – x 2 ) Reduce and you have m Step 3 - Find equation (y – y 1 ) = m(x – x 1 ) Step 4 - Check line Draw line and check b 23 ___-___ -34 - m = 1 7 y – 2 = 1 (x – – 3) 7 y = 1 x + 2 3 7 7 TOC 02/11/12 lntaylor ©

41. Now you try! TOC 02/11/12 lntaylor ©

42. 0,0 Find equation (3,2) (- 4,- 5) Step 1 – Identify information Start at 0,0; find first point Dot the first point Start at 0,0; find second point Put another dot Step 2 – Find the slope Find Δy (y 1 – y 2 ) Find Δx (x 1 – x 2 ) Reduce and you have m Step 3 - Find equation (y – y 1 ) = m(x – x 1 ) Step 4 - Check line Draw line and check b 2- 5 ___-___ 3- 4 - m = 1 y – 2 = 1(x - 3) y = x - 1 TOC 02/11/12 lntaylor ©

43. Now you try! (2,4) (3,-2) y = - 6x + 16 TOC 02/11/12 lntaylor ©

44. Determining Points on a Line TOC 02/11/12 lntaylor ©

45. 0,0 Is (2,3) on the line y = 2x + 1 Step 1 – Identify information Start at 0,0; find first point Dot the first point Use slope for 2 nd point Put another dot Draw line Step 2 – Find the point Start at 0,0 ; find given point Put a dot Determine if dot is on line Step 3 – Another way Substitute point into equation = yes ≠ no NO Is (2,3) on the line y = 2x + 1(2)3= 2+ 1 3 ≠ 5 the answer is NO TOC 02/11/12 lntaylor ©

46. Now you try! (2,3) y = 2x - 1 TOC 02/11/12 lntaylor ©

47. 0,0 Is (2,3) on the line y = 2x - 1 Step 1 – Identify information Start at 0,0; find first point Dot the first point Use slope for 2 nd point Put another dot Draw line Step 2 – Find the point Start at 0,0 ; find given point Put a dot Determine if dot is on line Step 3 – Another way Substitute point into equation = yes ≠ no YES Is (2,3) on the line y = 2x - 1(2)3= 2- 1 3 = 3 the answer is Yes TOC 02/11/12 lntaylor ©

48. Now you try! Is the point (-2,5) on the line y = 2x - 1 NO 5 ≠ - 5 TOC 02/11/12 lntaylor ©

49. Graphing Inequalities TOC 02/11/12 lntaylor ©

50. Graphing Inequalities Need to know the following –Boundary lines are either solid or dotted –≤ ≥ means solid line; point on line is included in the solution set –<> means dotted line; point on line is not included in solution set –Memorize the following rhyme Shoot an arrow Which line is right Turn back left And shade the night –What it means Shoot an arrow – means draw a line Which line is right – means solid (≤ ≥) or dotted ( ) Turn back left – means turn ≤≥<> back ¼ turn left And shade the night – means shade which way the arrow is pointing TOC 02/11/12 lntaylor ©

51. 0,0 Graph y ≥ 2x - 1 Step 1 – Identify information Start at 0,0; find first point Dot the first point Use slope for 2 nd point Put another dot Draw solid or dotted line Step 2 – Shade the graph Turn symbol 1/4 turn left Shade ≥ ≥ ≥ means solid TOC 02/11/12 lntaylor ©

52. Now you try! TOC 02/11/12 lntaylor ©

53. 0,0 Graph y < 2x - 1 Step 1 – Identify information Start at 0,0; find first point Dot the first point Use slope for 2 nd point Put another dot Draw solid or dotted line Step 2 – Shade the graph Turn symbol 1/4 turn left Shade < < < means dotted TOC 02/11/12 lntaylor ©

54. Now you try a hard one! TOC 02/11/12 lntaylor ©

55. 0,0 Graph y < 2 Step 1 – Identify information Start at 0,0; find first point Dot the first point No slope means no 2 nd point Draw solid or dotted line Step 2 – Shade the graph Turn symbol 1/4 turn left Shade < < < means dotted TOC 02/11/12 lntaylor ©