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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMarvin and Amelia each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Marvin has 1414 signatures, and Amelia has 1818. Marvin is collecting signatures at an average rate of 77 per day, whereas Amelia is averaging 33 signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many signatures will they both have? How long will that take?\newlineMarvin and Amelia will each have collected _\_ signatures in _\_ days.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineMarvin and Amelia each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Marvin has 1414 signatures, and Amelia has 1818. Marvin is collecting signatures at an average rate of 77 per day, whereas Amelia is averaging 33 signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many signatures will they both have? How long will that take?\newlineMarvin and Amelia will each have collected _\_ signatures in _\_ days.
  1. Define Variables: Define the variables for the number of signatures Marvin and Amelia will have after a certain number of days.\newlineLet's let MM represent the total number of signatures Marvin will have, and AA represent the total number of signatures Amelia will have. Let dd represent the number of days after which they will have the same number of signatures.
  2. Write Equations: Write the equations based on the given information.\newlineMarvin starts with 1414 signatures and collects 77 per day. Amelia starts with 1818 signatures and collects 33 per day. The system of equations that represents this situation is:\newlineM=14+7dM = 14 + 7d (Equation 11)\newlineA=18+3dA = 18 + 3d (Equation 22)\newlineWe want to find the value of dd for which MM equals 7700.
  3. Use Substitution: Use substitution to solve the system of equations.\newlineSince we are looking for the point where MM equals AA, we can set Equation 11 equal to Equation 22:\newline14+7d=18+3d14 + 7d = 18 + 3d
  4. Solve for d: Solve for d.\newlineSubtract 3d3d from both sides of the equation:\newline14+7d−3d=18+3d−3d14 + 7d - 3d = 18 + 3d - 3d\newline14+4d=1814 + 4d = 18\newlineSubtract 1414 from both sides:\newline4d=18−144d = 18 - 14\newline4d=44d = 4\newlineDivide both sides by 44:\newlined=4/4d = 4 / 4\newlined=1d = 1
  5. Calculate Signatures: Calculate the number of signatures each will have after dd days.\newlineSubstitute d=1d = 1 into either Equation 11 or Equation 22:\newlineUsing Equation 11 (Marvin's equation):\newlineM=14+7(1)M = 14 + 7(1)\newlineM=14+7M = 14 + 7\newlineM=21M = 21\newlineUsing Equation 22 (Amelia's equation):\newlineA=18+3(1)A = 18 + 3(1)\newlineA=18+3A = 18 + 3\newlineA=21A = 21
  6. Verify Solution: Verify that the solution makes sense in the context of the problem.\newlineBoth Marvin and Amelia will have 2121 signatures after 11 day, which is consistent with their rates of collecting signatures (77 per day for Marvin and 33 per day for Amelia). This means that the solution is reasonable.

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