Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Jon and Valeria 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, Jon has $28$ signatures, and Valeria has $10$ . Jon is collecting signatures at an average rate of $6$ per day, whereas Valeria is averaging $15$ signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have? $\newline$ In $\_$ days, Jon and Valeria will each have collected $\_$ signatures.

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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

Jon and Valeria 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, Jon has

28

signatures, and Valeria has

10

. Jon is collecting signatures at an average rate of

6

per day, whereas Valeria is averaging

15

signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have?

\newline

\_

days, Jon and Valeria will each have collected

\_

signatures.

Set Up Equations: Let $x$ represent the number of days and $y$ represent the number of signatures collected. We can set up two equations based on the given rates of signature collection for Jon and Valeria. $\newline$ For Jon: $\newline$ Starting signatures: $28$ $\newline$ Rate of collection: $6$ signatures per day $\newline$ Equation: $y = 6x + 28$ $\newline$ For Valeria: $\newline$ Starting signatures: $10$ $\newline$ Rate of collection: $15$ signatures per day $\newline$ Equation: $y = 15x + 10$ $\newline$ Now we have a system of equations: $\newline$ $1$ . $y = 6x + 28$ $\newline$ $2$ . $y = 15x + 10$
Set Equations Equal: To find the point where Jon and Valeria have the same number of signatures, we set the two equations equal to each other: $\newline$ $6x + 28 = 15x + 10$
Solve for x: Now we solve for x by subtracting $6x$ from both sides of the equation: $\newline$ $6x + 28 - 6x = 15x + 10 - 6x$ $\newline$ $28 = 9x + 10$
Isolate x: Next, we subtract $10$ from both sides to isolate the term with $x$ : $\newline$ $28 - 10 = 9x + 10 - 10$ $\newline$ $18 = 9x$
Find Value of $\newline$ $x$ : To find the value of $\newline$ $x$ , we divide both sides by $\newline$ $9$ : $\newline$ $\newline$ $\frac{18}{9} = \frac{9x}{9}$ $\newline$ $\newline$ $2 = x$ $\newline$ So, it will take $\newline$ $2$ days for Jon and Valeria to have the same number of signatures.
Substitute $x$ into Equation: Now we need to find the number of signatures $y$ that they will each have after $2$ days. We can substitute $x = 2$ into either of the original equations. We'll use Jon's equation: $\newline$ $y = 6x + 28$ $\newline$ $y = 6(2) + 28$ $\newline$ $y = 12 + 28$ $\newline$ $y = 40$ $\newline$ So, after $2$ days, Jon and Valeria will each have collected $40$ signatures.