Matrix word and elementary row reduction rules

In summary: That is to say, you can use some number of plan As, some number of plan Bs, and some number of plan Cs. So, let the number of plan As be A, the number of plan Bs be B, and the number of plan Cs be C. Then you want to solve this system of equations for A, B, and C:3A + 4B + 5C = 667A + 4B + 3C = 748A + 8B + 9C = 136In summary, this conversation is discussing a problem involving three different floor plans and determining if it is possible to use a certain number of each plan to reach a specific number of bedrooms.
  • #1
subopolois
86
0

Homework Statement


a house plan has 3 different floor plans:
Plan A- 3 three-bedroom units, 7 two-bedroom units, and 8 one-bedroom units
Plan B- 4 three- bedroom units, 4 two-bedroom units, and 8 one-bedroom units
Plan C- 5 three-bedroom units, 3 two-bedroom units, and 9 one bedroom units
is it possible to have 66 three- bedroom units, 74 two-bedroom units, 136 one-bedroom units

Homework Equations


all elementary row reduction rules

The Attempt at a Solution


so far I've put the above into a matrix
3 7 8|18
4 4 8|16
5 3 9|17
after do all elementary row operations this is my result
1 -3 0|-2
0 1 1/2|3/2
0 0 0|-5
now i know that if the result in the last row is what it is here it has no solution, but does this mean that the above problem is not possible? is there something more i have to do? and also in my final solution matrix correct?
 
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  • #2
I'm very surprised that you went from a word problem directly to an augmented matrix that supposedly represents a system of equations, apparently skipping the step of producing the system of equations. If you did, you didn't show the system or mention it.

The augmented matrix you show represents this system:
3x + 7y + 8z = 18
4x + 4y + 8z = 16
5x + 3y + 9z = 17

What exactly do x, y, and z represent? If this system had been consistent and you had been able to solve it, what would have x, y, and z represented? I don't mean their numeric values.

Where did you get the constants in the last column of the augmented matrix? Did you just add up the numbers in the row? That's what it looks like.

One of the questions you asked was whether your final solution matrix correct. A better question would have been, is my initial matrix correct?
Mark
 
  • #3
As it turns out, there are an infinite number of solutions for the system I'm working with, but only two of them are reasonable. I've checked them both and they give me the right number of one-, two-, and three-bedroom units, so I'm pretty confident I'm on the right track.
 
  • #4
Subopolois,
Hey, I wasn't trying to scare you away--I was trying to get you thinking before you started mechanically row-reducing an augmented matrix.

The problem is asking how many plan A floor plans and how many plan B floor plans and how many plan C floor plans can you use to come up with 66 3-BR apts, 74 2-BR apts, and 136 1-BR apts.
 

1. What is a matrix word?

A matrix word is a term used to describe a set of symbols or numbers arranged in a rectangular array, typically used to represent mathematical equations or data sets.

2. What are elementary row reduction rules?

Elementary row reduction rules are a set of operations that can be performed on a matrix to manipulate its rows and ultimately reduce it to its simplest form. These operations include multiplying a row by a constant, adding one row to another, and switching the positions of two rows.

3. Why is elementary row reduction useful?

Elementary row reduction is useful because it allows us to solve complex systems of equations or analyze large data sets more efficiently. By reducing a matrix to its simplest form, we can easily identify patterns and relationships within the data.

4. How do I perform elementary row reduction?

To perform elementary row reduction, you must first choose a target element (typically a pivot) in the matrix and use the elementary row operations to manipulate the other elements in the same row. Repeat this process for each row until the entire matrix is in its simplest form.

5. What are some common mistakes to avoid when performing elementary row reduction?

Some common mistakes to avoid when performing elementary row reduction include forgetting to perform the same operation on every row, making errors when multiplying or adding rows, and forgetting to update the target element after performing an operation.

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