Find the intersection of the following planes:
2x+y+6z=7
3x+4y+3z=-8
x-2y-4z=9
you have been in this website since 2005 :0
solve the system of equations for x, y, and z
those are the coordinates of the point of intersection
To find the intersection of the given planes, we need to solve the system of linear equations formed by the equations of the planes.
Let's represent the system of equations as a matrix equation:
A * X = B
Where A is the coefficient matrix, X is the column matrix of variables (x, y, and z), and B is the column matrix of constants.
The coefficient matrix (A) contains the coefficients of x, y, and z from the equations of the planes, respectively.
A = [[2, 1, 6], [3, 4, 3], [1, -2, -4]]
The column matrix of variables (X) contains x, y, and z, respectively.
X = [[x], [y], [z]]
The column matrix of constants (B) contains the constants on the right-hand side of the equations of the planes, respectively.
B = [[7], [-8], [9]]
To solve the system, we can use the matrix inverse.
The solution is given by:
X = A^(-1) * B
To find the inverse of matrix A, we can use various techniques like row reduction, cofactor expansion, or using matrix software.
Alternatively, if you have access to a matrix calculator or software, you can directly enter the matrices A and B into the calculator to find the solution.
Now, solving for X, we get the values of x, y, and z which represent the intersection point of the planes.