How many swimmers finished in first place and in third place?

Question

The Posted results of a recent high school swim meet state that had 24 individuals placed, earning a combined total of 53 points. First place earned 3 points, second place earned 2 points. and third place earned 1 point. there were as many first place finishers as second- and third-place finishers combined. Write a system of equations that represents how many people finished in each place.

Answer 1

Let x=the number of swimmers finished first place.

     y=the number of swimmers finished second place.

     z= the number of swimmers finished third place.

 

The first equation represents the total number of individuals who participated in the high school swim meet in which is:

  x+y+z=24

 

The second equation represents the combined total score in which is:

 3x+2y+z=53

 

Since the problem states that there were as many first place finisher as there were second place and third place finishers combined indicates x=y+z. This last equation is to be substituted into the first two equations; thereby, reducing them into two equations with two unknowns. Hence, the two equations are:

 

(y+z)+y+z=2y+2z=24

3(y+z)+2y+z=3y+3z+2y+z=5y+4z=53

 

Thus,

 

2y+2z=24

5y+4z=53

 

Next, multiply the first equation by -2 in which the system becomes:

 

-4y-4z=-48

 5y+4z=53

 

Add the two equations to eliminate the variable 'z' to reduce the two equations to one equation.

 

y=5

 

 2*5+2z=24

 10+2z=24

-10      -10

-------------

    2z=14  Divide both sides by 2 in which makes:

     z=7

 

Finally, x=y+z=5+7=12

 

Therefore, the following number of swimmers who finished first, second, and third: 1st) 12, 2nd) 5, and 3rd) 7.