Linear Algebra Problem Solving

Question

A class of 32 students was made up of people who were all 18, 19 and 20 years old. The average of their ages was 18.5. How many of each age bracket were in the class if the number of 18-year-olds was 6 more than the combined number of 19 and 20-year-olds? Find the equation and do the Gauss-Jordan reduction.

Answer 1

Let's turn these sentences into equations.

"A class of 32 students was made up of people who were all 18, 19 and 20 years old."

Let x represent the number of 18 year-olds, let y represent the number of 19 year-olds, and let z represent the number of 20 year-olds.

Then x + y + z = 32.

(Note: This is for the NUMBER OF STUDENTS of each age. I'm not using x, y, or z to represent age, but rather the number of students of each age.)

"The average of their ages was 18.5."

An average is the sum of observations (here, sum of students' ages) divided by the number of observations (here, 32). I don't know how many there are of each age of student, but their average age is 18.5. Now I can write an equation that explores AGE, not the number of students like in the previous equation.

(18x + 19y + 20z ) / 32 = 18.5

You could write this equation as (18/32)x + (19/32)y + (20/32)z = 18.5.

OR, to make it easier to work with, especially if this is being done by hand, let's get rid of the fraction by multiplying 32 on both sides so it is better suited for Gauss-Jordan elimination:

18x + 19y + 20z = 592

This means that all 32 students have a combined age of 592 years. Fun!

"...the number of 18-year-olds was 6 more than the combined number of 19 and 20-year-olds..."

Refer back to the variable definitions for x, y, and z.

x ---> "the number of 18 year-olds"

= ---> "was"

6 + ---> "6 more than"

(y + z) ---> "the combined number of 19 and 20 year-olds."

We get: x = 6 + (y + z). Re-arrange this to prepare it for Gauss-Jordan elimination:

x - y - z = 6

Your system should now look like this:

x + y + z = 32 (an equation discussing the count of each age of student)

18x + 19y + 20z = 592 (an equation discussing the combined age of the class, which we found from what was given about the average age of the class)

x - y - z = 6 (an equation that came from knowing that there were more members of one age than members of other ages)

This is your system. I'll leave you to do the Gauss-Jordan elimination. I hope this helps!