Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Band students are tested on, and required to pass, a certain number of scales during the year. As of today, Kendra has passed $2$ scales, whereas her friend Jackie has passed $15$ of them. Going forward, Kendra has committed to passing $4$ scales per week, and Jackie has committed to passing $3$ per week. At some point soon, the two friends will have passed the same number of scales. How long will that take? How many scales will that be? $\newline$ In $\underline{\quad}$ weeks, the friends will each have passed $\underline{\quad}$ scales.

Full solution

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

\newline

Band students are tested on, and required to pass, a certain number of scales during the year. As of today, Kendra has passed

2

scales, whereas her friend Jackie has passed

15

of them. Going forward, Kendra has committed to passing

4

scales per week, and Jackie has committed to passing

3

per week. At some point soon, the two friends will have passed the same number of scales. How long will that take? How many scales will that be?

\newline

\underline{\quad}

weeks, the friends will each have passed

\underline{\quad}

scales.

Define Variables: Define the variables for the number of scales Kendra and Jackie have passed. $\newline$ Let's let $K$ represent the total number of scales Kendra has passed after a certain number of weeks, and $J$ represent the total number of scales Jackie has passed after the same number of weeks.
Write Equations: Write the equations based on the given information. $\newline$ Kendra starts with $2$ scales and passes $4$ scales per week. So, $K = 4w + 2$ , where $w$ is the number of weeks. $\newline$ Jackie starts with $15$ scales and passes $3$ scales per week. So, $J = 3w + 15$ .
Set Equations Equal: Set the equations equal to each other to find when Kendra and Jackie will have passed the same number of scales. $4w + 2 = 3w + 15$
Solve for w: Solve for w by isolating the variable on one side of the equation. $\newline$ $4w - 3w = 15 - 2$ $\newline$ $w = 13$
Total Scales Kendra: Determine the total number of scales passed by Kendra after $w$ weeks. $\newline$ $K = 4w + 2$ $\newline$ $K = 4(13) + 2$ $\newline$ $K = 52 + 2$ $\newline$ K = $54$
Verify Jackie's Scales: Verify the result by checking if Jackie has also passed the same number of scales after $w$ weeks.$\newline$$J = 3w + 15$$\newline$$J = 3(13) + 15$$\newline$$J = 39 + 15$$\newline$J = $54$
Conclude Solution: Conclude the solution by stating how many weeks it will take and how many scales each will have passed. $\newline$ In $13$ weeks, the friends will each have passed $54$ scales.