Consider the system below.
{4x-2y=-12}
{3x-y=-3}
Solve the system by using a matrix equation.
+4 -2 -12
+3 -1 -3
gauss jordan
see
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
I appreciate the help thank you, Damon
x = 3, y = 12
Step 2: Enter the values for your matrix, and press the "Solve" button.
Divide row1 by 4
1 -1/2 -3
3 -1 -3
Add (-3 * row1) to row2
1 -1/2 -3
0 1/2 6
Divide row2 by 1/2
1 -1/2 -3
0 1 12
Add (1/2 * row2) to row1
1 0 3
0 1 12
You are welcome.
To solve the system using a matrix equation, we first need to write the given system of equations in matrix form.
Let's define the coefficient matrix A, the variable matrix X, and the constant matrix B as follows:
A = [[4, -2], [3, -1]]
X = [[x], [y]]
B = [[-12], [-3]]
Now, we can rewrite the given system of equations in matrix form as:
AX = B
Multiplying both sides of the equation by the inverse of A, we can isolate the variable matrix X:
X = A^(-1) * B
To solve the system, we need to find the inverse of matrix A. To calculate the inverse of a 2x2 matrix, follow these steps:
1. Swap the positions of the diagonal elements: [[-1, -2], [3, 4]]
2. Change the signs of the off-diagonal elements: [[-1, 2], [-3, 4]]
3. Multiply each element by the reciprocal of the determinant of the original matrix. The determinant of A is 4*(-1) - (-2)*(3) = 4 + 6 = 10. The reciprocal of 10 is 1/10. So, the inverse of matrix A is [[-1/10, 2/10], [-3/10, 4/10]] or [[-0.1, 0.2], [-0.3, 0.4]].
Now, substitute the values of A^(-1) and B in the equation X = A^(-1) * B:
X = [[-0.1, 0.2], [-0.3, 0.4]] * [[-12], [-3]]
Multiply the matrices to obtain the value of X:
X = [[-0.1 * -12 + 0.2 * -3], [-0.3 * -12 + 0.4 * -3]]
X = [[1.2 - 0.6], [3.6 - 1.2]]
X = [[0.6], [2.4]]
Therefore, the solution to the system of equations is x = 0.6 and y = 2.4.