Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. Ivan is trying to incorporate more exercise into his busy schedule. He has several short exercise routines he can complete at home. Last week, he worked out for a total of 102 minutes by doing 1 arm routine and 3 abdominal routines. This week, he has completed 1 arm routine and 4 abdominal routines and spent a total of 132 minutes exercising. How long does each routine last? An arm routine takes _____ minutes to complete, and an abdominal routine takes _____ minutes to complete.

Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.  Ivan is trying to incorporate more exercise into his busy schedule. He has several short exercise routines he can complete at home. Last week, he worked out for a total of 102 minutes by doing 1 arm routine and  3 abdominal routines. This week, he has completed 1 arm routine and  4 abdominal routines and spent a total of 132 minutes exercising. How long does each routine last?  An arm routine takes _____ minutes to complete, and an abdominal routine takes _____ minutes to complete.

Bytelearn · Accepted Answer

Define Variables: Let's define two variables: let a be the time it takes to complete an arm routine, and let b be the time it takes to complete an abdominal routine. We can then write two equations based on the information given:  1. Last week's workout: 1a + 3b = 102 minutes 2. This week's workout: 1a + 4b = 132 minutes  We can use these two equations to form a system of equations.Form System of Equations: To solve the system using elimination, we want to eliminate one of the variables. We can subtract the first equation from the second equation to eliminate variable a.  (1a + 4b) - (1a + 3b) = 132 - 102  This simplifies to:  1a - 1a + 4b - 3b = 132 - 102 0a + b = 30  So, b = 30 minutes.Elimination Method: Now that we know the value of b, we can substitute it back into one of the original equations to find the value of a. Let's use the first equation:  1a + 3(30) = 102  This simplifies to:  1a + 90 = 102  Now, we subtract 90 from both sides to solve for a:  1a = 102 - 90 a = 12  So, an arm routine takes 12 minutes to complete.Substitute and Solve: We have found that an arm routine takes 12 minutes (a = 12) and an abdominal routine takes 30 minutes (b = 30). We can check our work by plugging these values back into the second equation:  1(12) + 4(30) = 132  This simplifies to:  12 + 120 = 132  Which is a true statement, confirming our solution is correct.

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