A mixture of 5 pounds of fertilizer A, 13 pounds of fertilizer B, and 4 pounds of fertilizer C provides the optimal nutrients for a plant.

- Commercial brand X contains equal parts of fertilizer B and fertilizer C.
- Commercial brand Y contains one part of fertilizer A and two parts of fertilizer B.
- Commercial brand Z contains two parts of fertilizer A, five parts of fertilizer B, and two parts of fertilizer C.

How much of each fertilizer brand is needed to obtain the desired mixture?

(If the system is dependent, enter $ a $ for brand X, and express brand Y and brand Z in terms of $ a $.)

To answer the student's question, we need to set up and solve a system of linear equations that reflect the desired mix of fertilizer contents and the proportions found in each brand. The student learns to use algebraic methods to balance the components of the mixture correctly.

Explanation:

The student is asking how to combine different commercial fertilizer brands to create a desired mixture ratio for optimal plant nutrients. To solve the problem, we need to set up a system of linear equations based on the proportions of fertilizer A, B, and C in each commercial brand.

Let's assume we need to mix x pounds of brand X, y pounds of brand Y, and z pounds of brand Z to obtain the desired mixture of 5 pounds of fertilizer A, 13 pounds of fertilizer B, and 4 pounds of fertilizer C.

Brand compositions:

Brand X: B = C

Brand Y: A = 1 part, B = 2 parts

Brand Z: A = 2 parts, B = 5 parts, C = 2 parts

Equations based on the composition:

A: 0*X + 1/3*Y + 2/9*Z = 5

B: 1/2*X + 2/3*Y + 5/9*Z = 13

C: 1/2*X + 0*Y + 2/9*Z = 4

By solving these equations together, we can find the values for x, y, and z. However, if the system of equations is dependent, we may need to express one or two variables in terms of the third variable.

This problem requires a clear understanding of complete fertilizers, incomplete fertilizers, and their guaranteed analysis.