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anonymous

  • one year ago

its 2 30 AM need to wake up in 3 hours, please help me finish this last question fast A system of equations is shown below: -3x + 7y = -16 -9x + 5y = 16 Part A: Create an equivalent system of equations by replacing one equation with the sum of that equation and a multiple of the other. Show the steps to do this. (6 points) Part B: Show that the equivalent system has the same solution as the original system of equations. (4 points)

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  1. Michele_Laino
    • one year ago
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    if we multiply the first equation by -3, we get: \[9x - 21y = 48\]

  2. Michele_Laino
    • one year ago
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    Now if we sum that equation with the second one, what do you get?

  3. anonymous
    • one year ago
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    x-16y=64?

  4. Michele_Laino
    • one year ago
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    hint: \[9x - 21y + \left( { - 9x + 5y} \right) = 48 + 16\] please simplify

  5. anonymous
    • one year ago
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    but wouldnt that be x-16y=16?

  6. Michele_Laino
    • one year ago
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    no, since the next step is: \[9x - 21y - 9x + 5y = 48 + 16\] please simplify

  7. anonymous
    • one year ago
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    oh you put addition at first, x-25y=64

  8. Michele_Laino
    • one year ago
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    hint: what is 9 -9 =...

  9. anonymous
    • one year ago
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    0

  10. Michele_Laino
    • one year ago
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    ok! so the coefficient of the term with x is zero. Now what is -21 + 5= ...?

  11. anonymous
    • one year ago
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    -16

  12. Michele_Laino
    • one year ago
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    ok! so the coefficient of the term with y is -16

  13. Michele_Laino
    • one year ago
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    then we can write: \[0x - 16y = 64\] and finally: \[ - 16y = 64\]

  14. Michele_Laino
    • one year ago
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    am I right?

  15. anonymous
    • one year ago
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    yeah

  16. Michele_Laino
    • one year ago
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    ok! so the requested system of part A, can be this: \[\left\{ \begin{gathered} - 3x + 7y = - 16 \hfill \\ - 16y = 64 \hfill \\ \end{gathered} \right.\]

  17. anonymous
    • one year ago
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    thats it for part A?

  18. Michele_Laino
    • one year ago
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    yes!

  19. anonymous
    • one year ago
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    perfect, now what do we do for part B?

  20. Michele_Laino
    • one year ago
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    we can consider the last system, namely the equivalent system, and we can solve the second equation for y, namely: \[ - 16y = 64\] what is y?

  21. Michele_Laino
    • one year ago
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    hint: divide both sides by -16, what do you get?

  22. anonymous
    • one year ago
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    y=-4

  23. Michele_Laino
    • one year ago
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    ok!

  24. Michele_Laino
    • one year ago
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    now substitute that value of y into the first equation of the equivalent system, namely: \[ - 3x + 7y = - 16\] what equation do you get?

  25. anonymous
    • one year ago
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    -3x+7-4=16

  26. Michele_Laino
    • one year ago
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    not exactly, here is the right step: \[ - 3x + 7 \times \left( { - 4} \right) = - 16\] please simplify

  27. Michele_Laino
    • one year ago
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    what is: \[7 \times \left( { - 4} \right) = ...?\]

  28. anonymous
    • one year ago
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    -3x-21=-16?

  29. Michele_Laino
    • one year ago
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    hint: \[7 \times \left( { - 4} \right) = - 28\] am I right?

  30. anonymous
    • one year ago
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    oops lol, yeah you're right, i accidentally multiplied by -3

  31. anonymous
    • one year ago
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    sorry its real late over here, cant think properly

  32. Michele_Laino
    • one year ago
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    ok! So we have: \[ - 3x - 28 = - 16\]

  33. Michele_Laino
    • one year ago
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    now if we add 28 to both sides, we get: \[ - 3x - 28 + 28 = - 16 + 28\] please simplify

  34. anonymous
    • one year ago
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    -3x=12

  35. Michele_Laino
    • one year ago
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    ok! then if we divide both sides by -3, we can write: \[\frac{{ - 3x}}{{ - 3}} = \frac{{12}}{{ - 3}}\] please simplify

  36. anonymous
    • one year ago
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    -4

  37. anonymous
    • one year ago
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    x=-4

  38. Michele_Laino
    • one year ago
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    perfect! So the solution of the equivalent system is: \[\left\{ \begin{gathered} x = - 4 \hfill \\ y = - 4 \hfill \\ \end{gathered} \right.\]

  39. Michele_Laino
    • one year ago
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    now, please substitute that solution into the first equation of the original system, namely: \[ - 3x + 7y = - 16\] what do you get?

  40. anonymous
    • one year ago
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    -3-4+7-4=-16

  41. anonymous
    • one year ago
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    oops

  42. Michele_Laino
    • one year ago
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    hint: \[ - 3 \times \left( { - 4} \right) + 7 \times \left( { - 4} \right) = - 16\]

  43. anonymous
    • one year ago
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    gotta multiply

  44. Michele_Laino
    • one year ago
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    yes!

  45. anonymous
    • one year ago
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    yeah i meant that

  46. Michele_Laino
    • one year ago
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    what is (-3) * (-4)=... and 7*(-4)=...?

  47. anonymous
    • one year ago
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    12 -28

  48. Michele_Laino
    • one year ago
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    ok! so we can write: \[12 - 28 = - 16\]

  49. Michele_Laino
    • one year ago
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    now what is 12-28=...?

  50. anonymous
    • one year ago
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    -16

  51. Michele_Laino
    • one year ago
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    ok! So we can write: \[ - 16 = - 16\]

  52. anonymous
    • one year ago
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    ok thanks a lot! appreciate every thing, you taught me better than any teacher has in school!

  53. Michele_Laino
    • one year ago
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    reassuming the solution of the equivalent system makes the first equation an identity

  54. Michele_Laino
    • one year ago
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    now we have to do the same procedure for the second equation of the original system. If the solution of the equivalent system makes an identity the second equation of the original system then we can state that the two systems, namely the original system and the equivalent system have the same solution

  55. anonymous
    • one year ago
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    i think we only have to do it with one equation, but i might be wrong

  56. Michele_Laino
    • one year ago
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    so if we substitute the solution of the equivalent system: \[\left\{ \begin{gathered} x = - 4 \hfill \\ y = - 4 \hfill \\ \end{gathered} \right.\] into the second equation of the original system: \[ - 9x + 5y = 16\] what do you get?

  57. anonymous
    • one year ago
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    -9(-4)+5(-4)=16

  58. Michele_Laino
    • one year ago
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    ok! \[ - 9 \times \left( { - 4} \right) + 5 \times \left( { - 4} \right) = 16\]

  59. Michele_Laino
    • one year ago
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    now please simplify: what is (-9)*(-4)=...? and 5*(-4)=...?

  60. anonymous
    • one year ago
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    36 -20

  61. Michele_Laino
    • one year ago
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    so we can write: \[36 - 20 = 16\] now what is 36-20=...?

  62. anonymous
    • one year ago
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    16

  63. anonymous
    • one year ago
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    16=16

  64. Michele_Laino
    • one year ago
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    ok! so we got an identity again!

  65. Michele_Laino
    • one year ago
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    then we can state that both systems have the same solution, which is: \[\left\{ \begin{gathered} x = - 4 \hfill \\ y = - 4 \hfill \\ \end{gathered} \right.\]

  66. anonymous
    • one year ago
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    ok, thanks for all the help! greatly appreciate it

  67. Michele_Laino
    • one year ago
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    :)

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