Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Marvin 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 $14$ signatures, and Amelia has $18$ . Marvin is collecting signatures at an average rate of $7$ per day, whereas Amelia is averaging $3$ 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? $\newline$ Marvin and Amelia will each have collected $\_$ signatures in $\_$ days.

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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

Marvin 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

14

signatures, and Amelia has

18

. Marvin is collecting signatures at an average rate of

7

per day, whereas Amelia is averaging

3

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?

\newline

Marvin and Amelia will each have collected

\_

signatures in

\_

days.

Define Variables: Define the variables for the number of signatures Marvin and Amelia will have after a certain number of days. $\newline$ Let's let $M$ represent the total number of signatures Marvin will have, and $A$ represent the total number of signatures Amelia will have. Let $d$ represent the number of days after which they will have the same number of signatures.
Write Equations: Write the equations based on the given information. $\newline$ Marvin starts with $14$ signatures and collects $7$ per day. Amelia starts with $18$ signatures and collects $3$ per day. The system of equations that represents this situation is: $\newline$ $M = 14 + 7d$ (Equation $1$ ) $\newline$ $A = 18 + 3d$ (Equation $2$ ) $\newline$ We want to find the value of $d$ for which $M$ equals $7$ $0$ .
Use Substitution: Use substitution to solve the system of equations. $\newline$ Since we are looking for the point where $M$ equals $A$ , we can set Equation $1$ equal to Equation $2$ : $\newline$ $14 + 7d = 18 + 3d$
Solve for d: Solve for d. $\newline$ Subtract $3d$ from both sides of the equation: $\newline$ $14 + 7d - 3d = 18 + 3d - 3d$ $\newline$ $14 + 4d = 18$ $\newline$ Subtract $14$ from both sides: $\newline$ $4d = 18 - 14$ $\newline$ $4d = 4$ $\newline$ Divide both sides by $4$ : $\newline$ $d = 4 / 4$ $\newline$ $d = 1$
Calculate Signatures: Calculate the number of signatures each will have after $d$ days. $\newline$ Substitute $d = 1$ into either Equation $1$ or Equation $2$ : $\newline$ Using Equation $1$ (Marvin's equation): $\newline$ $M = 14 + 7(1)$ $\newline$ $M = 14 + 7$ $\newline$ $M = 21$ $\newline$ Using Equation $2$ (Amelia's equation): $\newline$ $A = 18 + 3(1)$ $\newline$ $A = 18 + 3$ $\newline$ $A = 21$
Verify Solution: Verify that the solution makes sense in the context of the problem. $\newline$ Both Marvin and Amelia will have $21$ signatures after $1$ day, which is consistent with their rates of collecting signatures ( $7$ per day for Marvin and $3$ per day for Amelia). This means that the solution is reasonable.