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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.When Colin does push-ups and sit-ups, it takes a total of seconds. In comparison, he needs seconds to do push-ups and sit-ups. How long does it take Colin to do each kind of exercise?It takes Colin seconds to do a push-up and seconds to do a sit-up.
Full solution
Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.When Colin does push-ups and sit-ups, it takes a total of seconds. In comparison, he needs seconds to do push-ups and sit-ups. How long does it take Colin to do each kind of exercise?It takes Colin seconds to do a push-up and seconds to do a sit-up.
- Define Variables: Let's denote the time it takes Colin to do one push-up as seconds and the time to do one sit-up as seconds. The first situation gives us the equation .
- Form Equations: The second situation gives us the equation . We can simplify this equation by dividing all terms by , which gives us .
- Solve System: Now we have a system of two equations:) ) We can solve this system using substitution or elimination. Let's use substitution since the second equation is already solved for .
- Substitute and Simplify: From the second equation, we can express in terms of : . Now we can substitute this expression for into the first equation.
- Distribute and Combine: Substituting into the first equation gives us . Now we need to distribute and solve for .
- Solve for x: Distributing gives us . Combining like terms, we get .
- Find : Subtracting from both sides gives us . Dividing both sides by gives us .
- Final Results: Now that we have , we can substitute it back into the equation to find . So , which gives us .
- Final Results: Now that we have , we can substitute it back into the equation to find . So , which gives us .We have found that and . This means it takes Colin seconds to do a push-up and second to do a sit-up.
