Solve the equation 2x + 5y - 3z = -1. Write the general solution as a matrix equation.
[x] [ ] [1]
[y]= [ ] X [s]
[z] [ ] [t]
I'm supposed to fill in that blank in the middle, but I don't understand how to solve this if I'm only given one equation. please explain
[x] [...] [1]
[y]= [...] X [s]
[z] [...] [t]
to align it better I put dots in the blanks...and I think s and t are free variables that be be any real number
okay i put it wrong I'm sorry the question said
Convert to the Alternate form (Exponential INTO logarithm)
(2 5 -3) (x y z)T = -1
The transpose is for readability, since it has to be a column vector.
I still don't really understand how to do it...
I did some guessing and ended up getting this much:
[x] [-1/2 _ _] [1]
[y]= [0 1 0] X [s]
[z] [0 0 1] [t]
however I still can't figure out the last two values in the first row.
To solve the equation 2x + 5y - 3z = -1, we need to find the general solution in terms of x, y, and z. However, with only one equation, we cannot determine the values of x, y, and z independently.
Instead, we can rewrite the equation in matrix form to find the general solution. Let's denote the column vector [x, y, z] as [X], and the matrix on the left side of the equation as A:
A = [2, 5, -3]
Then, we can rewrite the equation as:
AX = [-1]
To solve for [X], we can perform matrix algebra by multiplying both sides of the equation by the inverse of matrix A:
A^(-1) * AX = A^(-1) * [-1]
Since A is a 1x3 matrix, we can treat it as a row vector. Finding the inverse of such a matrix is not possible because it does not satisfy the necessary conditions. Thus, we cannot directly multiply both sides by the inverse and solve for [X].
Therefore, the equation 2x + 5y - 3z = -1 does not have a unique solution. Instead, it represents a plane in 3-dimensional space, and any point on that plane satisfies the equation. The general solution can be expressed parametrically:
[x] [1] [x]
[y] = [s] --> [y] = [y]
[z] [t] [z]
Here, [s] and [t] are any real numbers that can be chosen freely, and [x, y, z] represents any point on the plane that satisfies 2x + 5y - 3z = -1.