1 / 115

Solving Systems of Equations

Solving Systems of Equations. Graphing. There are three methods to solving systems of equations by graphing: Write both equations in slope – intercept form and graph Write both equations in slope-intercept form and graph using the calculator Solve for the x and y intercepts of each equation.

abraham-clay
Télécharger la présentation

Solving Systems of Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Solving Systems of Equations

  2. Graphing There are three methods to solving systems of equations by graphing: • Write both equations in slope – intercept form and graph • Write both equations in slope-intercept form and graph using the calculator • Solve for the x and y intercepts of each equation

  3. Graphing Solve the following system of equations by graphing: -6x +3y = -15 -4x +y = -11

  4. Graphing – Method 1 Write both equations in slope – intercept form and graph. To do this, solve each equation for y -6x +3y = -15 -4x +y = -11

  5. Graphing – Method 1 Writing -6x +3y = -15 in slope intercept form: -6x +3y = -15 +6x +6x 3y = 6x – 15 • 3 3 y = 2x - 5

  6. Graphing – Method 1 Writing -4x + y = -11 in slope intercept form: -4x + y = -11 +4x +4x y = 4x - 11

  7. Graphing – Method 1 Graph both equations using their slope and y –intercept by starting at the y-intercept and using their slope to do rise over run. Equation 1: y = 2x - 5 y intercept is (0, -5) slope is rise 2, run 1 Equation 2: y = 4x – 11 y intercept is (0, -11) slope is rise 4, run 1

  8. Graphing – Method 1 The lines intersect at the point (1,3)

  9. Graphing – Method 2 Write both equations in slope-intercept form and graph using the calculator -6x +3y = -15 -4x +y = -11 The equations were already solved for slope-intercept form in method 1, so: y = 2x – 5 y = 4x – 11

  10. Graphing – Method 2 (TI-84+) • Turn the calculator on • Hit the “Y=” key • Type in the first equation next to Y1 • Use the “X,T,O,n” key to type “X” • Hit “Enter” • Type in the second equation next to Y2 • Hit “Enter”

  11. Graphing – Method 2 (TI-84+) 8) Hit the graph button to see the graph

  12. Graphing – Method 2 (TI-84+) 9) If necessary, adjust the graph by changing the zoom You can zoom in, or out by hitting the zoom button and then selecting option 2 or 3. Once selected, press enter again when you see the graph Zoom standard goes back to the regular zoom

  13. Graphing – Method 2 (TI-84+) 10) When looking at the graph hit the “CALC” button. Do this by hitting the “2ND” key followed by the “TRACE” key 11) Move down to choice five and select “intersect” 12) Press “Enter” and the calculator will return to the graph.

  14. Graphing – Method 2 (TI-84+) 13) The calculator will prompt you to select the first curve. Use the arrows to put the blinking cursor on one of the lines 14) Hit “Enter”

  15. Graphing – Method 2 (TI-84+) 15) The calculator will prompt you to select the second curve. Use the arrows to put the blinking cursor on the other line (the calculator should have already done this for you) 16) Hit “Enter”

  16. Graphing – Method 2 (TI-84+) 17) The calculator will prompt you to guess the location of the intersection. Use the arrow keys to move the flashing curser close to the intersection 18) Hit “Enter”

  17. Graphing – Method 2 (TI-84+) 19) The calculator will then tell you the intersection. In this case, “X=3, Y=1” 20) Write your answer as an ordered pair (3,1)

  18. Graphing – Method 2 (TI-89) • Turn the calculator on • Hit the “Y=” key by hitting Diamond + F1 • Type in the first equation next to Y1 • Hit “Enter” • Type in the second equation next to Y2 • Hit “Enter”

  19. Graphing – Method 2 (TI-89) 7) Hit the graph button to see the graph - Do this by hitting diamond and then F3

  20. Graphing – Method 2 (TI-89) 8) If necessary, adjust the graph by changing the zoom You can zoom in, or out by hitting the zoom button (F2) and then selecting option 2 or 3. Once selected, press enter again when you see the graph Zoom standard (option 6) goes back to the regular zoom

  21. Graphing – Method 2 (TI-89) 9) When looking at the graph select the “Math” menu. Do this by hitting the “F5” key 10) Move down to choice five and select “intersection” 11) Press “Enter” and the calculator will return to the graph.

  22. Graphing – Method 2 (TI-89) 12) The calculator will prompt you to select the first curve. Use the arrows to put the blinking cursor on one of the lines 13) Hit “Enter”

  23. Graphing – Method 2 (TI-89) 14) The calculator will prompt you to select the second curve. Use the arrows to put the blinking cursor on the other line (the calculator should have already done this for you) 15) Hit “Enter”

  24. Graphing – Method 2 (TI-89) 16) The calculator will prompt you to select the lower bound of the intersection. Use the arrow keys to move below or to the left of the intersection 17) Hit “Enter”

  25. Graphing – Method 2 (TI-89) 18) The calculator will prompt you to select the upper bound of the intersection. Use the arrow keys to move above or to the right of the intersection 19) Hit Enter

  26. Graphing – Method 2 (TI-89) 20) The calculator will then tell you the intersection. In this case, “X=3, Y=1” 21) Write your answer as an ordered pair (3,1)

  27. Graphing – Method 3 Graph by solving for the x and y intercepts of each equation: -6x +3y = -9 -4x +y = -8

  28. Graphing – Method 3 Find the x and y intercepts of the first equation: -6x +3y = -15 y-intercept, let x=0 -6x +3y = -15 -6(0) + 3y = -15 3y = -15 • 3 y = -5 y-int = (0,-5) x-intercept, let y=0 -6x +3y = -15 -6x +3(0) = -15 -6x = -15 -6 -6 x = -15/-6 = 5/2 x-int = (5/2,0)

  29. Graphing – Method 3 Find the x and y intercepts of the second equation: -4x +y = -11 y-intercept, let x=0 -4x +y = -11 -4(0) + y = -11 y = -11 y-int = (0,-11) x-intercept, let y=0 -4x +y = -11 -4x +(0) = -11 -4x = -11 -4 -4 x = -11/-4 = 11/4 x-int = (11/4,0)

  30. Graphing – Method 3 • Graph by plotting the x and y intercepts of each line and connecting them to form the line • The solution is the intersection: the point (3,1)

  31. Substitution 1) Solve one of the equations for a variable 2) Substitute the solved equation into the OTHER equation in place of the variable you solved for 3) Solve the new equation for the remaining variable. 4) Once you find an answer for the first variable, substitute that answer into one of the original two equations and solve for the second variable. 5) Write your answer as an ordered pair.

  32. Substitution ** If both variables cancel out, the lines are either parallel, or they are the same line • If the lines are parallel then there is no solution, and when the equation is solved, it will result in an answer that is NOT true like 0 = 4 • If the lines are the same, then there is an infinite number of solutions, and when the equation is solved, it will result in an answer that IS true like 0=0 or 5=5.

  33. Substitution – Example 1 Solve the following two equations using substitution: y = 2x + 1 3x – 2y = -4

  34. Substitution – Example 1 • Solve one of the equations for a variable - The first equation is already solved for y: y = 2x + 1

  35. Substitution – Example 1 2) Substitute the solved equation into the OTHER equation in place of the variable you solved for So, substitute y = 2x + 1 into 3x – 2y = -4 3x – 2(2x + 1) = -4

  36. Substitution – Example 1 3) Solve the new equation for the remaining variable. 3x – 2(2x + 1) = -4 3x -2(2x) -2(1) = -4 3x – 4x – 2 = -4 -x – 2 = -4 +2 +2 -x = -2 -1 -1 x = 2

  37. Substitution – Example 1 4) Once you find an answer for the first variable, substitute that answer into one of the original two equations and solve for the second variable. y = 2x + 1 and x = 2, so y = 2(2) + 1 y = 4 + 1 y = 5

  38. Substitution – Example 1 5) Write your answer as an ordered pair. x = 2 and y = 5 so the answer is: (2,5)

  39. Substitution – Example 1 To check your work, substitute your answer into all of the original equations: Substitute in (2, 5) for x, y y = 2x + 1 5 = 2(2) + 1 5 = 4+1 5 = 5 Both equations balance, so our answer is a solution 3x – 2y = -4 3(2) – 2(5) = -4 6 – 10 = -4 -4 = -4

  40. Substitution – Example 2 Solve the following two equations using substitution: 3x - 2y = 5 4x + 4y = 20

  41. Substitution – Example 2 • Solve one of the equations for a variable - Solving the second equation for y is the easiest: 4x + 4y = 20 -4x -4x 4y = 20 – 4x • 4 4 y = 5 - x

  42. Substitution – Example 2 2) Substitute the solved equation into the OTHER equation in place of the variable you solved for So, substitute y = 5 – x into 3x - 2y = 5 3x – 2(5-x) = 5

  43. Substitution – Example 2 3) Solve the new equation for the remaining variable. 3x – 2(5-x) = 5 3x - 2(5) - 2(-x) = 5 3x - 10 + 2x = 5 5x – 10 = 5 +10 +10 5x = 15 • 5 x = 3

  44. Substitution – Example 2 4) Once you find an answer for the first variable, substitute that answer into one of the original two equations and solve for the second variable. - Substitute x = 3 into 3x - 2y = 5 3(3) – 2y = 5 9 – 2y = 5 -9 -9 -2y = -4 -2 -2 y = 2

  45. Substitution – Example 2 5) Write your answer as an ordered pair. x = 3 and y = 2 so the ordered pair is: (3,2)

  46. Substitution – Example 3 Solve the following two equations using substitution: -2x + 4y = -12 -x + 2y = 2

  47. Substitution – Example 3 • Solve one of the equations for a variable - Solving the second equation for x is very easy: -x + 2y = 2 -2y -2y -x = 2 – 2y -1 -1 -1 x = -2 + 2y

  48. Substitution – Example 3 2) Substitute the solved equation into the OTHER equation in place of the variable you solved for So, substitute x= -2 + 2y into -2x + 4y = -12 -2(-2+2y) +4y = -12

  49. Substitution – Example 3 3) Solve the new equation for the remaining variable. -2(-2+2y) +4y = -12 -2(-2) + -2(2y) + 4y = -12 4 – 4y + 4y = -12 4 = -12 ** Both variables are eliminated, and 4 = -12 is not true. Thus the lines are parallel and the answer is “no solution”

  50. Substitution – Example 4 Solve the following two equations using substitution: -2x + y = 3 6x + -3y = -9

More Related