Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Students in Mrs. Russell's third grade class are working on times tables, and they demonstrate mastery by passing tests. Madelyn has passed $1$ test so far. Her classmate, Quinn, has passed $7$ tests of them. From now on, Madelyn plans to take and pass $2$ tests per week. Meanwhile, Quinn plans to do $1$ per week. At some point, Madelyn will catch up to Quinn. How long will it take? How many tests will each child have passed? $\newline$ In $\_$ weeks, the children will each have passed $\_$ tests.

Full solution

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

\newline

Students in Mrs. Russell's third grade class are working on times tables, and they demonstrate mastery by passing tests. Madelyn has passed

1

test so far. Her classmate, Quinn, has passed

7

tests of them. From now on, Madelyn plans to take and pass

2

tests per week. Meanwhile, Quinn plans to do

1

per week. At some point, Madelyn will catch up to Quinn. How long will it take? How many tests will each child have passed?

\newline

\_

weeks, the children will each have passed

\_

tests.

Define variables: Let's define the variables for the number of tests Madelyn and Quinn will have passed after a certain number of weeks. Let $M$ represent the total number of tests Madelyn has passed and $Q$ represent the total number of tests Quinn has passed. We know that initially, Madelyn has passed $1$ test and Quinn has passed $7$ tests.
Equation for Madelyn: We can write the first equation for Madelyn, who plans to pass $2$ tests per week. So, after $w$ weeks, the total number of tests Madelyn will have passed is $M = 1 + 2w$ .
Equation for Quinn: We can write the second equation for Quinn, who plans to pass $1$ test per week. So, after $w$ weeks, the total number of tests Quinn will have passed is $Q = 7 + w$ .
Set Madelyn equal to Quinn: Since we are looking for the point where Madelyn catches up to Quinn, we set $M$ equal to $Q$ . This gives us the equation $1 + 2w = 7 + w$ .
Solve for w: To solve for w, we subtract $w$ from both sides of the equation to get $1 + w = 7$ . Then we subtract $1$ from both sides to find $w = 6$ .