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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineStudents in Mrs. Russell's third grade class are working on times tables, and they demonstrate mastery by passing tests. Madelyn has passed 11 test so far. Her classmate, Quinn, has passed 77 tests of them. From now on, Madelyn plans to take and pass 22 tests per week. Meanwhile, Quinn plans to do 11 per week. At some point, Madelyn will catch up to Quinn. How long will it take? How many tests will each child have passed?\newlineIn _\_ weeks, the children will each have passed _\_ tests.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineStudents in Mrs. Russell's third grade class are working on times tables, and they demonstrate mastery by passing tests. Madelyn has passed 11 test so far. Her classmate, Quinn, has passed 77 tests of them. From now on, Madelyn plans to take and pass 22 tests per week. Meanwhile, Quinn plans to do 11 per week. At some point, Madelyn will catch up to Quinn. How long will it take? How many tests will each child have passed?\newlineIn _\_ weeks, the children will each have passed _\_ tests.
  1. 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 MM represent the total number of tests Madelyn has passed and QQ represent the total number of tests Quinn has passed. We know that initially, Madelyn has passed 11 test and Quinn has passed 77 tests.
  2. Equation for Madelyn: We can write the first equation for Madelyn, who plans to pass 22 tests per week. So, after ww weeks, the total number of tests Madelyn will have passed is M=1+2wM = 1 + 2w.
  3. Equation for Quinn: We can write the second equation for Quinn, who plans to pass 11 test per week. So, after ww weeks, the total number of tests Quinn will have passed is Q=7+wQ = 7 + w.
  4. Set Madelyn equal to Quinn: Since we are looking for the point where Madelyn catches up to Quinn, we set MM equal to QQ. This gives us the equation 1+2w=7+w1 + 2w = 7 + w.
  5. Solve for w: To solve for w, we subtract ww from both sides of the equation to get 1+w=71 + w = 7. Then we subtract 11 from both sides to find w=6w = 6.

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