Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Neil 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 $96$ minutes by doing $2$ arm routines and $3$ abdominal routines. This week, he has completed $2$ arm routines and $2$ abdominal routines and spent a total of $72$ minutes exercising. How long does each routine last? $\newline$ An arm routine takes $\_$ minutes to complete, and an abdominal routine takes $\_$ 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

Neil 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

96

minutes by doing

2

arm routines and

3

abdominal routines. This week, he has completed

2

arm routines and

2

abdominal routines and spent a total of

72

minutes exercising. How long does each routine last?

\newline

An arm routine takes

\_

minutes to complete, and an abdominal routine takes

\_

minutes to complete.

Define variables: Let's define two variables: let $a$ be the time in minutes it takes to complete an arm routine, and $b$ be the time in minutes it takes to complete an abdominal routine. We can then write two equations based on the information given: $\newline$ $1$ . For last week's workouts: $2a + 3b = 96$ (since Neil did $2$ arm routines and $3$ abdominal routines for a total of $96$ minutes). $\newline$ $2$ . For this week's workouts: $2a + 2b = 72$ (since Neil did $2$ arm routines and $2$ abdominal routines for a total of $72$ minutes). $\newline$ We will use these two equations to form a system of equations.
Write equations: To solve the system using elimination, we want to eliminate one of the variables. We can do this by multiplying the second equation by $1$ . $5$ to make the coefficient of $b$ in the second equation equal to the coefficient of $b$ in the first equation: $\newline$ $1.5 \times (2a + 2b) = 1.5 \times 72$ $\newline$ This gives us: $\newline$ $3a + 3b = 108$ $\newline$ Now we have a new system of equations: $\newline$ $1$ . $2a + 3b = 96$ $\newline$ $2$ . $3a + 3b = 108$
Use elimination method: Next, we subtract the first equation from the second equation to eliminate $b$ : $\newline$ $(3a + 3b) - (2a + 3b) = 108 - 96$ $\newline$ This simplifies to: $\newline$ $a = 12$ $\newline$ So, an arm routine takes $12$ minutes to complete.
Subtract equations: Now that we know the value of $a$ , we can substitute it back into one of the original equations to find $b$ . Let's use the first equation: $\newline$ $2(12) + 3b = 96$ $\newline$ This simplifies to: $\newline$ $24 + 3b = 96$ $\newline$ Now we solve for $b$ : $\newline$ $3b = 96 - 24$ $\newline$ $3b = 72$ $\newline$ $b = 24$ $\newline$ So, an abdominal routine takes $24$ minutes to complete.