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?

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Mathematics

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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 :

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LuisJ7162go LuisJ7162go · 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

[tex]\begin{cases}18x+11y=76 \\ 7x+19y=59\end{cases}[/tex]

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

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

Then,

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

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

[tex]\begin{gathered} y=2 \\ \Rightarrow18x+11y=18x+22 \\ \Rightarrow18x+22=76 \\ \Rightarrow18x=54 \\ \Rightarrow x=3 \end{gathered}[/tex]

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