Solve the equation 2x + 5y - 3z = -1. Write the general solution as a matrix equation.
I did some guessing and ended up getting this much:
[x]= [-1/2 _ _ ] X [1]
[y]= [ 0 1 0 ] X [s]
[z]= [ 0 0 1 ] X [t]
however I still can't figure out the last two values in the first row
To find the general solution for the equation 2x + 5y - 3z = -1, we need to rewrite it in matrix form.
The equation can be written as:
[2 5 -3] [x y z] = [-1]
To solve for the variables x, y, and z, let's denote the matrix of variables as [X] and the solution matrix as [B].
So, [2 5 -3] [X] = [-1]
To find the general solution, we need to find the inverse of the coefficient matrix [2 5 -3], denoted as [A].
[A] = [2 5 -3]⁻¹
To find the inverse of a matrix, we can use the following steps:
1. Calculate the determinant of the matrix [2 5 -3].
Determinant, denoted as |A|, is calculated as:
|A| = (2)(1) - (5)(-3) = 2 + 15 = 17
2. Swap the diagonals of the coefficient matrix [2 5 -3].
[2 5 -3] becomes [17 -5 -3].
3. Change the sign of the off-diagonals of the coefficient matrix.
[17 -5 -3] becomes [17 5 3].
4. Multiply the resulting matrix by 1/determinant.
[17 5 3] * (1/17) = [1 5/17 3/17].
Now, we have the inverse of the coefficient matrix, [A] = [1 5/17 3/17].
To find the general solution, multiply the inverse [A] by the solution matrix [B] = [ -1 ].
[X] = [A] * [B]
[X] = [1 5/17 3/17] * [-1]
[X] = [-1, -5/17, -3/17]
So, the general solution for the equation 2x + 5y - 3z = -1 is:
x = -1,
y = -5/17,
z = -3/17.
Hence, the general solution can be written as a matrix equation:
[x y z] = [-1, -5/17, -3/17]