Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Students in Mrs. Ortega's third grade class are working on times tables, and they demonstrate mastery by passing tests. Trent has passed $12$ tests so far. His classmate, Gordon, has passed $3$ tests of them. From now on, Trent plans to take and pass $2$ tests per week. Meanwhile, Gordon plans to do $5$ per week. At some point, Trent will catch up to Gordon. How many tests will each child have passed? How long will it take? $\newline$ Trent and Gordon will each have passed $\_\_\_$ tests in $\_\_\_$ weeks.

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Algebra 1

Solve a system of equations using any method: word problems

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

\newline

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

12

tests so far. His classmate, Gordon, has passed

3

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

2

tests per week. Meanwhile, Gordon plans to do

5

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

\newline

Trent and Gordon will each have passed

\_\_\_

tests in

\_\_\_

weeks.

Define Variables: Define the variables for the number of tests passed by Trent and Gordon. $\newline$ Let $T$ represent the total number of tests passed by Trent after a certain number of weeks. $\newline$ Let $G$ represent the total number of tests passed by Gordon after the same number of weeks. $\newline$ Let $w$ represent the number of weeks that have passed.
Set Equations: Set up the equations based on the given information. $\newline$ Trent has already passed $12$ tests and plans to pass $2$ more tests per week. So, $T = 12 + 2w$ . $\newline$ Gordon has already passed $3$ tests and plans to pass $5$ more tests per week. So, $G = 3 + 5w$ .
Equation Solution: Since Trent will catch up to Gordon, at some point $T$ will equal $G$ . Therefore, we have the equation: $12 + 2w = 3 + 5w$ .
Calculate Tests Passed: Solve the equation for $w$ to find out after how many weeks Trent will catch up to Gordon. $12 + 2w = 3 + 5w$ $2w - 5w = 3 - 12$ $-3w = -9$ $w = -9 / -3$ $w = 3$
Verify Equality: Calculate the total number of tests passed by Trent and Gordon after $w$ weeks. $\newline$ Since $w = 3$ , we substitute this value into the equations for $T$ and $G$ . $\newline$ For Trent: $T = 12 + 2(3) = 12 + 6 = 18$ $\newline$ For Gordon: $G = 3 + 5(3) = 3 + 15 = 18$
Verify Equality: Calculate the total number of tests passed by Trent and Gordon after $w$ weeks. $\newline$ Since $w = 3$ , we substitute this value into the equations for $T$ and $G$ . $\newline$ For Trent: $T = 12 + 2(3) = 12 + 6 = 18$ $\newline$ For Gordon: $G = 3 + 5(3) = 3 + 15 = 18$ Verify that the values of $T$ and $G$ are equal, as they should be when Trent catches up to Gordon. $\newline$ $T = 18$ $\newline$ $G = 18$ $\newline$ Since $T$ equals $G$ , the solution is correct.