Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Janelle and Bridget 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, Janelle has $15$ signatures, but Bridget just started and doesn't have any yet. Janelle is collecting signatures at an average rate of $5$ per hour, while Bridget can get $20$ 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 $\underline{\hspace{2cm}}$ hours, Janelle and Bridget will each have collected $\underline{\hspace{2cm}}$ .

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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

Janelle and Bridget 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, Janelle has

15

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

5

per hour, while Bridget can get

20

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

\underline{\hspace{2cm}}

hours, Janelle and Bridget will each have collected

\underline{\hspace{2cm}}

Define Variables: Let's define the variables: $\newline$ Let $J$ represent the total number of signatures Janelle will have. $\newline$ Let $B$ represent the total number of signatures Bridget 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 Janelle: $J = 15 + 5h$ (since she starts with $15$ and collects $5$ per hour) $\newline$ For Bridget: $B = 20h$ (since she starts with $0$ and collects $20$ per hour)
Set Equations Equal: Since we are looking for the point where Janelle and Bridget have the same number of signatures, we set the two equations equal to each other: $\newline$ $15 + 5h = 20h$
Solve for $h$ : Now we solve for $h$ by subtracting $5h$ from both sides of the equation: $\newline$ $15 + 5h - 5h = 20h - 5h$ $\newline$ $15 = 15h$
Divide and Solve: Next, we divide both sides by $15$ to solve for $h$ : $\frac{15}{15} = \frac{15h}{15}$ $1 = h$
Substitute and Find: Now that we know $h = 1$ , we can find out how many signatures they will each have by substituting $h$ back into either J's or B's equation. Let's use Janelle's equation: $\newline$ $J = 15 + 5(1)$ $\newline$ $J = 15 + 5$ $\newline$ $J = 20$
Final Signatures: Since we know that Janelle and Bridget will have the same number of signatures, Bridget will also have $20$ signatures after $1$ hour.