Chapter 3.4, Problem 20PPS

(a)

To determine

System of three equations representing the number of people finished in each place.

Answer to Problem 20PPS

System of three equations:

  $x+y+z=24$

And

  $3x+2y+z=53$

And

  $x=y+z$

Explanation of Solution

Given information:

In a high school swim meet,

24 individuals were placed.

Total 53 points were earned.

Such that

1^st place earned 3 points

2^nd place earned 2 points

3^rd place earned 1 point

Let

x people finished 1^st

And

y people finished 2^nd

And

z people finished 3^rd

Now,

The total number of people placed is the sum of number of people placed at each position.

We know that

24 individuals were placed.

Thus,

  $x+y+z=24$ … (1)

We also know

Total 53 points were earned.

Since we do not know how many people are there at each position.

Then

Write 3x for 1^st place, 2y for 2^nd place, z for 3^rd place.

Thus,

  $3x+2y+z=53$ … (2)

Also,

According to the statement,

There were as many 1^st placed finishers as 2^nd and 3^rd placed finishers combined.

Thus,

  $x=y+z$ … (3)

Therefore,

The system of equations we have

  $x+y+z=24$

And

  $3x+2y+z=53$

And

  $x=y+z$

(b)

To determine

Swimmers finished in 1^st place, in 2^nd place, and in 3^rd place.

Answer to Problem 20PPS

12 swimmers finished in 1^st place.

5 swimmers finished in 2^nd place.

7 swimmers finished in 3^rd place.

Explanation of Solution

Given information:

In a high school swim meet,

24 individuals were placed.

Total 53 points were earned.

Such that

1^st place earned 3 points

2^nd place earned 2 points

3^rd place earned 1 point

The original system of equation:

  $x+y+z=24$ … (1)

And

  $3x+2y+z=53$ … (2)

And

  $x=y+z$ … (3)

Substitute Eq. (3) into Eq. (1),

The new equation becomes

  $2x=24$

Divide both sides by 2:

  $x=\frac{24}{2}=12$

Now,

Subtract Eq. (1) from Eq. (2), z will be eliminated.

The new equation becomes

  $2x+y=29$

Substitute the value of x :

  $2\left(12\right)+y=29$

Simplify:

  $24+y=29$

Subtract 24 from both sides:

  $y=29-24=5$

Now,

Substitute the values of x and y into Eq. (3):

The new equation becomes

  $12=5+z$

Rewrite the equation:

  $5+z=12$

Subtract 5 from both sides:

  $z=12-5=7$

Therefore,

12 swimmers finished in 1^st place, 5 swimmers finished in 2^nd place, and 7 swimmers finished in 3^rd place.

(c)

To determine

Discuss the false e-mailed statement and unreasonable solution.

Answer to Problem 20PPS

47 points cannot be scored because the number of people at 2^nd place cannot be -1.

Explanation of Solution

Given information:

In a high school swim meet,

24 individuals were placed.

According to the e-mailed statement,

Total 47 points were earned instead of 53.

Such that

1^st place earned 3 points

2^nd place earned 2 points

3^rd place earned 1 point

According to the e-mailed statement,

Total 47 points were earned instead of 53.

Then

Eq. (2) becomes Eq. (2’)

  $3x+2y+z=47$ … (2’)

Now,

In the system of equations,

Eq. (2) is replaced by Eq. (2’).

Thus,

  $x+y+z=24$ … (1)

And

  $3x+2y+z=47$ … (2’)

And

  $x=y+z$ … (3)

Substitute Eq. (3) into Eq. (1),

The new equation becomes

  $2x=24$

Divide both sides by 2:

  $x=\frac{24}{2}=12$

Now,

Substitute the value of x in (2’):

  $3\left(12\right)+2y+z=47$

Simplify:

  $36+2y+z=47$

Subtract 36 from both sides:

  $2y+z=11$

Rewrite the equation:

  $y+\left[y+z\right]=11$

Substitute Eq. (3):

  $y+x=11$

Substitute the value of x :

  $y+12=11$

Subtract 12 from both sides:

  $y=11-12=-1$

Since the number of people at 2^nd place cannot be -1.

Therefore,

It is not possible that 47 points were scored.