Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ 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? $\newline$ An arm routine takes $\underline{\hspace{3em}}$ minutes to complete, and an abdominal routine takes $\underline{\hspace{3em}}$ minutes to complete.

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Grade 8

Solve a system of equations using elimination: word problems

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

\newline

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?

\newline

An arm routine takes

\underline{\hspace{3em}}

minutes to complete, and an abdominal routine takes

\underline{\hspace{3em}}

minutes to complete.

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: $\newline$ $1$ . Last week's workout: $1a + 3b = 102$ minutes $\newline$ $2$ . This week's workout: $1a + 4b = 132$ minutes $\newline$ 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 $a$ . $\newline$ $(1a + 4b) - (1a + 3b) = 132 - 102$ $\newline$ This simplifies to: $\newline$ $1a - 1a + 4b - 3b = 132 - 102$ $\newline$ $0a + b = 30$ $\newline$ 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: $\newline$ $1a + 3(30) = 102$ $\newline$ This simplifies to: $\newline$ $1a + 90 = 102$ $\newline$ Now, we subtract $90$ from both sides to solve for $a$ : $\newline$ $1a = 102 - 90$ $\newline$ $a = 12$ $\newline$ 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: $\newline$ $1(12) + 4(30) = 132$ $\newline$ This simplifies to: $\newline$ $12 + 120 = 132$ $\newline$ Which is a true statement, confirming our solution is correct.