Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Janet and Kenny 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, Janet has $27$ signatures, but Kenny just started and doesn't have any yet. Janet is collecting signatures at an average rate of $7$ per hour, while Kenny can get $10$ 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 signatures will they both have? How many hours will have gone by? $\newline$ Janet and Kenny will each have collected $\_$ in $\_$ hours.

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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

Janet and Kenny 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, Janet has

27

signatures, but Kenny just started and doesn't have any yet. Janet is collecting signatures at an average rate of

7

per hour, while Kenny can get

10

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 signatures will they both have? How many hours will have gone by?

\newline

Janet and Kenny will each have collected

\_

\_

hours.

Define Variables: Let's define the variables: $\newline$ Let $J$ represent the total number of signatures Janet will have. $\newline$ Let $K$ represent the total number of signatures Kenny will have. $\newline$ Let $h$ represent the number of hours that have gone by.
Write Equations: We can write two equations to represent the situation: $\newline$ For Janet: $J = 27 + 7h$ (since she starts with $27$ and collects $7$ per hour) $\newline$ For Kenny: $K = 10h$ (since he starts with $0$ and collects $10$ per hour)
Set Equal: Since we are looking for the point where they have the same number of signatures, we set $J$ equal to $K$ : $27 + 7h = 10h$
Solve for $h$ : Now we solve for $h$ by subtracting $7h$ from both sides of the equation: $\newline$ $27 + 7h - 7h = 10h - 7h$ $\newline$ $27 = 3h$
Find h Value: Divide both sides by $3$ to find the value of h: $\newline$ $27 \div 3 = 3h \div 3$ $\newline$ $9 = h$
Substitute $h$ : Now that we have the number of hours, we can find the number of signatures they will each have by substituting $h$ back into either $J$ or $K$ . We'll use $J$ : $J = 27 + 7h$ $J = 27 + 7(9)$
Calculate Janet's Signatures: Calculate the total number of signatures for Janet: $J = 27 + 63$ $J = 90$
Calculate Kenny's Signatures: Since $J = K$ , Kenny will also have $90$ signatures after $9$ hours.