The discussion revolves around the task of creating a four-dimensional system of equations that has a specific solution, namely (-2, 5, -6, 1). Participants explore different approaches to formulate such a system, considering both simplicity and the potential use of matrices.
Participants generally agree on the basic approach to creating the system, but there is no consensus on the preferred format or notation for presenting the solution.
The discussion reflects varying interpretations of what constitutes a "system" and how it should be represented, indicating potential dependencies on specific educational contexts.
sawdee said:I know how to solve systems, but the question asks to work backwards.
"Create a 4-d system that has a solution (-2,5,-6,1)."
How would i do this?
I like Serena said:Hi sawdee! Welcome to MHB! (Smile)
Without any other conditions, we can go for the simplest system that satisfies your requirement.
We can pick the system:
\begin{cases}
x_1=-2 \\
x_2 = 5 \\
x_3 = -6 \\
x_4 = 1
\end{cases}
No point in making it more complicated unless there are more restrictions.
We're talking about math - the simplest solution possible should be considered the best one.
sawdee said:Thank you! :) Happy to be here.
I picked that solution as well, but in terms of format I'm not sure how to set up the answer. Should I use a matrix with those 4 as my x values, and leave everything else as 0?
I think this is exactly what he wanted, thank you!I like Serena said:It depends a bit on what your teacher considers to be a "system".
If it's supposed to involve a matrix, we can pick the identity matrix with right hand values that correspond to the solution.
That is:
$$\begin{bmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
x_3 \\
x_4
\end{bmatrix} =
\begin{bmatrix}
-2 \\
5 \\
-6 \\
1
\end{bmatrix}$$
It's really just the same thing as the "system" I suggested though.
The only difference being what we pick as the notation.