Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.When Tanvi does 18 push-ups and 11 sit-ups, it takes a total of 76 seconds. In comparison, she needs 59 seconds to do 7 push-ups and 19 sit-ups. How long does it take Tanvi to do each kind of exercise?

Answer 1

Let x and y the time it takes Tanvi to do a push-up and a sit-up, respectively; thus, the system of equations is

`\begin{cases}18x+11y=76 \\ 7x+19y=59\end{cases}`

To solve it using elimination, multiply the second equation by 18/7 and subtract the result to the first equation, as shown below

`\begin{gathered} (18)/(7)(7x+19y)=(18)/(7)\cdot59 \\ \Rightarrow18x+(18)/(7)\cdot19y=(18)/(7)\cdot59 \\ \Rightarrow18x+(342)/(7)y=(1062)/(7) \end{gathered}`

Then,

`\begin{gathered} 18x+11y-(18x+(342)/(7)y)=76-(1062)/(7) \\ \Rightarrow(11-(342)/(7))y=76-(1062)/(7) \\ \Rightarrow y=(76-(1062)/(7))/((11-(342)/(7))) \\ \Rightarrow y=2 \end{gathered}`

Substitute the value of y=2 into the first equation,

`\begin{gathered} y=2 \\ \Rightarrow18x+11y=18x+22 \\ \Rightarrow18x+22=76 \\ \Rightarrow18x=54 \\ \Rightarrow x=3 \end{gathered}`

Thus, the answer is x=3, y=2. 3 seconds per push-up and 2 seconds per sit-up