2. Writing Linear Equations By Lindsay Hojnowski (2014) Buffalo State College 04/2014L. Hojnowski © 20141 Click here to play tutorial introduction Slope-Intercept Form Point-Slope Form Standard (General) Form

3. Learning Objectives The learner will be able to write a linear equation given a point and the slope with 85% degree of accuracy using slope-intercept form. The learner will be able to write the linear equations given two points with 85% degree of accuracy using slope-intercept form. Students will be able to write a linear equation in point-slope form given a point and the slope 90% of the time. 04/2014L. Hojnowski © 20142 Aim for the Target

4. Learning Objectives Continued… Given an equation in point-slope form, students will be able to change the equation into slope-intercept form 80% of the time. Given an equation in point-slope form, students will be able to change the equation into standard form 80% of the time. 04/2014L. Hojnowski © 20143 Aim for the Target

5. Menu 04/2014L. Hojnowski © 2014 Slope-Intercept Form- Given the slope and a point Slope-Intercept Form Example 2 Slope-Intercept Form Example 3 and 4 4 Slope-Intercept Form Example 1 References Slope-Intercept Form- Given two points Given two points- Example 1 Given two points- Example 2 Given two points- Example 3 Point-Slope Form Point-Slope Form Example 1 and 2 Point-Slope  Standard Form Example 1 Point-Slope  Standard Form Example 2 Point-Slope  Slope- Intercept Form Example 1 Point-Slope  Slope- Intercept Form Example 2 Quiz Question #1 Quiz Question #2 Quiz Question #3 Quiz Question #6 Quiz Question #4 Quiz Question #5 Quiz Question #7

6. Writing Equations in Slope-Intercept Form Given the slope and a point STEPS: 1)Identify x, y, and m 2)Plug these values into y = mx + b 3)Solve for b 4)Write the answer in slope-intercept form (y = mx+b) 04/2014L. Hojnowski © 20145 Slope-intercept form

7. Given a Point and Slope Example 1 04/2014L. Hojnowski © 20146 Steps to writing linear equations STEPS: 1)Identify x, y, and m 2)Plug these values into y = mx + b 3)Solve for b 4)Write the answer in slope-intercept form (y = mx+b) Example 1: Write the equation of the line that passes through (2, 1) with a slope of 3 1) x = 2, y = 1 and m = 33) 1 = 6 + b 2)y = mx + b -6 -6 1 = 3(2)+ b -5 = b 4) y = mx + b y = 3x - 5

8. Given a Point and Slope Example 2 04/2014L. Hojnowski © 20147 STEPS: 1)Identify x, y, and m 2)Plug these values into y = mx + b 3)Solve for b 4)Write the answer in slope-intercept form (y = mx+b) Example 2: Write the equation of the line that passes through (5, -2) with a slope of 1/3. 1)x = 5, y = -2 and m = 1/33) -2 = 1 + b 2)y = mx + b -1 -1 -2 = 3(1/3)+ b -3 = b 4) y = mx + b y = 1/3x - 3 Y- intercept and slope

9. Given a Point and Slope Example 3 and 4 Example 3: (8, 3) and m = 1 STEP 1: x = 8, y = 3, m = 1 STEP 2: y = mx + b 3 = 1(8) + b 3 = 8 + b STEP 3: -8 -8 -5 = b STEP 4: y = mx + b y = x - 5 04/2014L. Hojnowski © 20148 Example 4: (-6, 7) and m = 0 STEP 1: x = -6, y = 7, m = 0 STEP 2: y = mx + b 7 = 0(-6) + b 7 = 0 + b STEP 3: **Anything times zero is zero** 7 = b STEP 4: y = mx + b y = 7 Horizontal line

10. Writing Equations in Slope-Intercept Form Given two points 04/2014L. Hojnowski © 20149 STEPS: 1)Use the slope formula to find m 2)Choose one of the given points (the easier one) and plug x, y, and m into y = mx + b 3)Solve for b 4)Write the answer in slope-intercept form (y = mx+b) Two Points

11. Given Two Points Example 1 04/2014L. Hojnowski © 201410 STEPS: 1)Use the slope formula to find m 2)Choose one of the given points (the easier one) and plug x, y, and m into y = mx + b 3)Solve for b 4)Write the answer in slope-intercept form (y = mx+b) Example 1: Write the equation of the line that passes through (3, 1) and (2, 4) 1)m = y 2 – y 1 2) (3, 1) 3) 1 = -9 + b x 2 – x 1 y = mx + b +9 +9 1 = -3 (3) + b 10 = b m = 4 – 1 2 – 3 4) y = mx + b m = -3 y = -3x +10

12. Given Two Points Example 2 04/2014L. Hojnowski © 201411 STEPS: 1)Use the slope formula to find m 2)Choose one of the given points (the easier one) and plug x, y, and m into y = mx + b 3)Solve for b 4)Write the answer in slope-intercept form (y = mx+b) Example 2: Write the equation of the line that passes through (-4, -2) and (-5, -6) 1)m = y 2 – y 1 2) (-4, -2) 3) -2 = -16+ b x 2 – x 1 y = mx + b +16 +16 -2 = 4(-4) + b 14 = b m = -6 – -2 -5 – -4 m = -4 4) y = mx + b -1 y = 4x + 14 m = 4

13. Given Two Points Example 3 04/2014L. Hojnowski © 201412 Example 3: Write the equation of the line that passes through (3, 5) and (-2, 2) 1)m = y 2 – y 1 2) (3, 5) 3) 5 = 9/5 + b x 2 – x 1 y = mx + b -9/5 -9/5 5 = 3/5(3) + b 16/5 = b m = 2 – 5 -2 – 3 m = -3 4) y = mx + b -5 y = 3/5x + 16/5 m = 3 5 Slop Formula

14. Point-Slope Form 04/2014L. Hojnowski © 201413 Point-Slope Formula: y- y 1 = m (x – x 1 ) – Use this formula when you are given a point and a slope – Also use this formula when the question tells you too (READ CAREFULLY!) – Click on the picture below to see a video on how to use the point-slope formula Point-Slope Formula

15. Point-Slope Form Example 1 and 2 04/2014L. Hojnowski © 201414 Example 1: Write an equation in point- slope from for the line that passes through the point (-2, 1) and has a slope of -6. y – y 1 = m (x – x 1 ) y – 1 = -6 (x - - 2) y – 1 = -6 (x + 2) Example 2: Write an equation in point- slope from for the line that passes through the point (-2, -8) and has a slope of 5/6. y – y 1 = m (x – x 1 ) y – -8 = 5/6 (x - - 2) y + 8 = 5/6 (x + 2) Point-Slope Formula

16. Point-Slope  Standard Form Example 1 04/2014L. Hojnowski © 201415 Example 1: Write y + 7 = 5 (x + 3) in standard form. y + 7 = 5 (x + 3) y + 7 = 5x + 15 Distribute -7 -7 Get y by itself (7s cancel) y = 5x + 8 Combine like terms -5x -5x Rewrite into standard form -5x + y = 8 A = -5, B = 1, C = 8 STEPS: 1)Distribute 2)Rewrite into standard form (Ax + By = C), identify A, B, and C

17. Point-Slope  Standard Form Example 2 04/2014L. Hojnowski © 201416 Example 2: Write y - 1= -2/3 (x - 5) in standard form. y - 1= -2/3 (x - 5) y - 1= (-2/3)x + 10/3 +1 +1 y = (-2/3)x + 13/3 +(2/3)x (2/3)x+ y = 13/3 3((2/3)x+ y = 13/3) 2x + 3y = 13 y – y 1 = m (x – x 1 )  Ax + By = C Point-Slope Form  Standard Form Distribute Get y by itself (1s cancel) Combine like terms Rewrite into standard form **Standard form can only contain integers (multiply the whole equation by 3)** A = 2, B = 3, C = 13

18. Point-Slope  Slope-Intercept Example 1 04/2014L. Hojnowski © 201417 Example 1: Write y + 3 = 3/2 (x + 1) in slope-intercept form. y + 3 = 3/2 (x + 1) y + 3 = (3/2)x + 3/2 -3 -3 y = (3/2)x – 3/2 Or y = 1.5x – 1.5 STEPS: 1)Distribute 2)Rewrite into slope-intercept form (y = mx + b) and identify m (slope) and b (y-intercept) Distribute Get y by itself (3s cancel) Combine like terms (slope-intercept form) m = 3/2, b = 3/2

19. Point-Slope  Slope-Intercept Example 2 04/2014L. Hojnowski © 201418 Example 2: Write y – 10 = 4 (x + 6) in slope-intercept form. y – 10 = 4 (x + 6) y – 10 = 4x + 24 +10 +10 y = 4x + 34 y – y 1 = m (x – x 1 )  Ax + By = C Point-Slope Form  Standard Form Distribute Get y by itself (10s cancel) Combine like terms (slope-intercept form) m = 4, b = 34

20. Quiz Question #1 04/2014L. Hojnowski © 201419 1.Which of the following formulas is slope-intercept form? a. a. y = mx +b b. y – y 1 = m (x – x 1 ) c. Ax + By = Cd. y = ax 2 + bx +c b. c. d.

21. Correct!! 04/2014L. Hojnowski © 201420 -Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept Quiz Question #1 Quiz Question #2 Smile Point-Slope Formula

22. Try Again… 04/2014L. Hojnowski © 201421 This is the point-slope formula. We use this formula when we are given a point and a slope or when the problem tells us too. Quiz Question #1 Quiz Question #2 Try Again Point-Slope Formula

23. Try Again… 04/2014L. Hojnowski © 201422 This is standard form of a linear equation. This form is also referred to as general form. Quiz Question #1 Quiz Question #2 Try Again Standard Form

24. Try Again… 04/2014L. Hojnowski © 201423 This is standard form of a quadratic function. You should be very familiar with this since it was in Chapter 9! Quiz Question #1 Quiz Question #2 Try Again Standard Form- Quadratic

25. Quiz Question # 2 04/2014L. Hojnowski © 201424 2. Write the equation of a line that passes through (2, 1) and has a slope of 4. Your final answer should be in slope-intercept form. a.a. y – 1 = 4 (x – 2) b. y = 4x – 7 c. -4x + y = -7 d. y = -4x + 7b. c. d.

26. Try Again… 04/2014L. Hojnowski © 201425 If the question asked you to write the equation of the line in point-slope form, you would be correct. From this answer, you need to distribute and get y by itself. Quiz Question #1 Quiz Question #2 Try Again Quiz Question #3

27. Correct!! 04/2014L. Hojnowski © 201426 You plugged in and solved correctly! Slope-intercept form is y = mx + b; by just knowing this detail, answers #1 and #3 could be eliminated. Smile Quiz Question #1 Quiz Question #2 Quiz Question #3

28. Try Again… 04/2014L. Hojnowski © 201427 Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 If the question asked you to write the equation of the line in standard, you would be correct. You went one step too far.

29. Try Again… 04/2014L. Hojnowski © 201428 Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 There is a sign off in your equation. This is in slope-intercept form though! You are right about that.

30. Quiz Question # 3 04/2014L. Hojnowski © 201429 3. Write the linear equation in point-slope form using the following information: (-7, 2), m = 6 a.a. y + 2 = 6 (x – 7)b. y - 2 = 6 (x + 7) c. -6x + y = 44 d. y = 6x + 44b. c. d.

31. Try Again… 04/2014L. Hojnowski © 201430 Point-slope form is y – y 1 = m (x – x 1 ), it is not a plus sign in between the y’s and x’s. You are correct that this is point-slope form. Quiz Question #2 Quiz Question #1 Try Again Quiz Question #4 Quiz Question #3

32. Correct!! 04/2014L. Hojnowski © 201431 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 You plugged in and solved correctly! Point-slope form is y – y 1 = m (x – x 1 ); by just knowing this detail, answers #3 and #4 could be eliminated.

33. Try Again… 04/2014L. Hojnowski © 201432 Try Again If the question asked you to write the equation of the line in standard form, you would be correct. You went a few steps too far. Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3

34. Try Again… 04/2014L. Hojnowski © 201433 Try Again If the question asked you to write the equation of the line in slope-intercept form, you would be correct. You did not need to distribute or get y alone. Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3

35. Quiz Question # 4 04/2014L. Hojnowski © 201434 4. Using slope-intercept form, write the equation passing through the points (-5, 3) and (0, -7). a. a. y = 2x + 7 b. y = 2x - 5c. y = -2x - 5d. y = -2x - 7b. c. d.

36. Try Again… 04/2014L. Hojnowski © 201435 Careful with your signs! -7 – 3 = -10 and 0 - -5 = 5. A - - (negative negative) makes it a positive! Try again using this knowledge. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5

37. Try Again… 04/2014L. Hojnowski © 201436 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Careful with your signs and solving! -7 – 3 = -10 and 0 - -5 = 5. A - - (negative negative) makes it a positive! Remember anything multiplied by zero will given you zero! Try again using this knowledge.

38. Try Again… 04/2014L. Hojnowski © 201437 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Remember anything multiplied by zero will given you zero! Try again using this knowledge.

39. Correct!! 04/2014L. Hojnowski © 201438 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 You plugged in and solved correctly! You found the correct slope and solved correctly to find b.

40. Quiz Question # 5 04/2014L. Hojnowski © 201439 5. Write y – 10 = 2 (x – 8) in standard form. a.a. 2x - y = 6 b. 2x + y = 6c. -2x + y = -6 d. y = 2x -6b. c. d.

41. Try Again… 04/2014L. Hojnowski © 201440 Careful when you want to get 2x to the other side, you have to subtract 2x not add 2x! Also, you did not move your y term so it should still be positive. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6

42. Try Again… 04/2014L. Hojnowski © 201441 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Careful when you want to get 2x to the other side, you have to subtract 2x not add 2x! Remember to also bring down the sign of your constant!

43. Correct!! 04/2014L. Hojnowski © 201442 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 You plugged in and solved correctly! Awesome job subtracting 2x and bringing down the sign with the constant (-6).

44. Try Again… 04/2014L. Hojnowski © 201443 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Careful the question wanted the equation in standard form. This equation is in slope-intercept form. You need to do one more step to get this equation in standard form.

45. Quiz Question # 6 6. Johnny wrote y + 4 = 2/3 (x + 7) in standard from as (-2/3)x + y = 2/3. Jenna wrote the same equation in standard form and got 2x – 3y = -2. Who is correct? a.a. Johnny b. Jenna b. 04/2014L. Hojnowski © 201444

46. Try Again… 04/2014L. Hojnowski © 201445 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Johnny solved correctly but he forget to do one step. Standard form can only contain integers (so you cannot have fractions). He would need to multiply the entire equation by -3.

47. Correct!! 04/2014L. Hojnowski © 201446 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Jenna remembered that standard form can only contain integers. She got rid of the fractions by multiplying the equation by -3.

48. Quiz Question # 7 7. Write y + 6 = -3/4 (x + 8) in slope-intercept form. a. a. y + 6 = (-3/4)x – 6 b. y = (-3/4)x – 12 c. y = (-3/4)x d. y = (-3/4)x + 12b. c. d. 04/2014L. Hojnowski © 201447

49. Try Again… 04/2014L. Hojnowski © 201448 This is one step after point-slope form. You need to do one more step to get to slope-intercept form. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7

50. Correct!! 04/2014L. Hojnowski © 201449 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 You solved correctly! Awesome job distributing, subtracting 6 from both sides and simplifying.

51. Try Again… 04/2014L. Hojnowski © 201450 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Careful when you want to get +6 to the other side, you have to subtract 6 not add 6!

52. Try Again… 04/2014L. Hojnowski © 201451 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Careful when you combine -6 and -6, they have the same sign. So you just add them and keep the negative sign.

53. References McGraw-Hill Companies. (2014). Glencoe Algebra 1 Common Core Edition. New York: McGraw Hill. Seminars.usb.ac.ir. (2011). Hitting the objectives, Retrieved on September 14 th, 2012, from http://www.teambuildinggames.org/role-of-the-team-building- facilitator. http://www.teambuildinggames.org/role-of-the-team-building- facilitator Smiley Face, Retrieved on September 14 th, 2012, from http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/images /happy-face1.png. http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/images /happy-face1.png Wee, E. (2011). Try again, Retrieved on September 15 th, 2012, from http://radionjournals.blogspot.com/2011/04/try-again-part-3-caring- for-children.html. http://radionjournals.blogspot.com/2011/04/try-again-part-3-caring- for-children.html 04/2014L. Hojnowski © 201452 Reference from the dictionary