Question

Topic: Reduced row echelon form
Thanks for the help in advance

Answer 1

Let f, s, and t denote the hours needed by the first, second, and third sellers, respectively, to sell a given plant.

"In one hour, the first grower sells 3 fly traps … The second grower sells 1 fly trap … The third grower sells 3 fly traps …" - this means that for each hour f, the first grower sells a total of 3f fly traps. Similarly, the second grower sells 1s fly traps, and the third grower sells 3t fly traps. Together, they want to sell 39 fly traps, so that

3f + s + 3t = 39

You set up two more equations for the sundews and bladderworts:

f + 2s + 2t = 30

2f + s + 2t = 28

Now write the system of equations in augmented matrix form and row reduce it:

`\left[\begin{array}c3&1&3&39\\1&2&2&30\\2&1&2&28\end{array}\right]`

Add row 1 to -3 (row 2); this is to say, treating each row as a list, multiply through the second row by -3, then add the first row to that result. For example,

(3, 1, 3, 39) + -3 (1, 2, 2, 30) = (3, 1, 3, 39) + (-3, -6, -6, -90)

… = (3 - 3, 1 - 6, 3 - 6, 39 - 90)

… = (0, -5, -3, -51)

Also, add 2 (row 1) to -3 (row 3) :

`\left[\begin{array}ccc3&1&3&39\\0&-5&-3&-51\\0&-1&0&-6\end{array}\right]`

Multiply through the last two rows by -1 to make everything positive, and swap them around:

`\left[\begin{array}ccc3&1&3&39\\0&1&0&6\\0&5&3&51\end{array}\right]`

Next, add -5 (row 2) to row 3 :

`\left[\begin{array}c3&1&3&39\\0&1&0&6\\0&0&3&21\end{array}\right]`

Multiply through row 3 by 1/3 :

`\left[\begin{array}ccc3&1&3&39\\0&1&0&6\\0&0&1&7\end{array}\right]`

Add -1 (row 2) and -3 (row 3) to row 1 :

`\left[\begin{array}c3&0&0&12\\0&1&0&6\\0&0&1&7\end{array}\right]`

Multiply through row 1 by 1/3 :

`\left[\begin{array}ccc1&0&0&4\\0&1&0&6\\0&0&1&7\end{array}\right]`

This row-reduced matrix tells you that f = 4, s = 6, and t = 7. In other words, in order for the three growers to collectively sell the required number of plants, the first seller needs to work for 4 hours, the second seller needs to work for 6 hours, and the third seller needs to work for 7 hours.