Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ 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? $\newline$ In $\_$ hours, Porter and Matt will each have collected $\_$ .

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

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?

\newline

\_

hours, Porter and Matt will each have collected

\_

Define Variables: Let's define the variables: $\newline$ Let $P$ be the total number of signatures Porter has, and $M$ be the total number of signatures Matt has. $\newline$ 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: $\newline$ For Porter: $P = 16 + 9t$ (since Porter starts with $16$ signatures and collects $9$ per hour) $\newline$ 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 $M$ : $16 + 9t = 17t$
Solve for $t$ : Now we solve for $t$ by subtracting $9t$ from both sides of the equation: $\newline$ $16 + 9t - 9t = 17t - 9t$ $\newline$ $16 = 8t$
Divide by $8$ : Next, we divide both sides by $8$ to solve for $t$ : $\frac{16}{8} = \frac{8t}{8}$ $t = 2$
Substitute $t$ 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 $M$ : $P = 16 + 9(2) = 16 + 18 = 34$
Substitute $t$ 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.