Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Porter and Matt both plan to run for a spot on the school board in their city. They must each collect a certain number of signatures to get their name on the ballot. So far, Porter has 16 signatures, but Matt just started and doesn't have any yet. Porter is collecting signatures at an average rate of 9 per hour, while Matt can get 17 signatures every hour. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many hours will have gone by? How many signatures will they both have? In _____ hours, Porter and Matt will each have collected _____.

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.  Porter and Matt both plan to run for a spot on the school board in their city. They must each collect a certain number of signatures to get their name on the ballot. So far, Porter has 16 signatures, but Matt just started and doesn't have any yet. Porter is collecting signatures at an average rate of 9 per hour, while Matt can get 17 signatures every hour. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many hours will have gone by? How many signatures will they both have?   In _____ hours, Porter and Matt will each have collected _____.

Bytelearn · Accepted Answer

Define Variables: Let's define the variables: Let P be the total number of signatures Porter has, and M be the total number of signatures Matt has. Let t be the number of hours that have gone by since Matt started collecting signatures.Write Equations: We can write two equations to represent the situation: For Porter: P = 16 + 9t (since Porter starts with 16 signatures and collects 9 per hour) For Matt: M = 0 + 17t (since Matt starts with 0 signatures and collects 17 per hour)Set Equal: Since we are looking for the point where they have the same number of signatures, we set P equal to M: 16 + 9t = 17tSolve for t: Now we solve for t by subtracting 9t from both sides of the equation: 16 + 9t - 9t = 17t - 9t 16 = 8tDivide by 8: Next, we divide both sides by 8 to solve for t: 16 ÷ 8 = 8t ÷ 8 t = 2Substitute t into P: Now that we know t = 2 hours, we can find out how many signatures they both have by substituting t back into either P or M: P = 16 + 9(2) = 16 + 18 = 34Substitute t into M: To check our work, we can also substitute t into M: M = 17(2) = 34 Since P and M are equal, our solution is correct.

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