Use Gauss-Jordan to solve
3x + 2y = -4
4x – 2y = -10
To solve the system of equations using the Gauss-Jordan method, we can first set up the augmented matrix. The augmented matrix is formed by taking the coefficients of the variables on the left side of the equations and the constants on the right side of the equations.
The given system of equations is:
3x + 2y = -4
4x – 2y = -10
To create the augmented matrix, we arrange the coefficients of the variables and the constants in a matrix:
| 3 2 | -4 |
| 4 -2 | -10 |
Now, we perform row operations to transform the matrix into row-echelon form and then further transform it into reduced row-echelon form.
The goal is to make the elements below and above each leading coefficient (the leftmost non-zero entry in a row) equal to zero using the following row operations:
1. Multiply a row by a non-zero constant (Rk → kRk).
2. Add or subtract a multiple of one row from another row (Rk → Rk + kRj).
Let's go through the steps to solve the given system of equations:
Step 1: Divide the first row by 3 to make the leading coefficient in the first row equal to 1.
| 1 2/3 | -4/3 |
| 4 -2 | -10 |
Step 2: Multiply the first row by -4 and add it to the second row to eliminate the leading coefficient in the second row.
| 1 2/3 | -4/3 |
| 0 -10/3 | 2/3 |
Step 3: Multiply the second row by -3/10 to make the leading coefficient in the second row equal to 1.
| 1 2/3 | -4/3 |
| 0 1 | -1/5 |
Step 4: Multiply the second row by -2/3 and subtract it from the first row to eliminate the remaining non-zero entry in the first row.
| 1 0 | -6/5 |
| 0 1 | -1/5 |
The matrix is now in reduced row-echelon form. The first column corresponds to x, and the second column corresponds to y.
From the reduced row-echelon form, we can read the solution to the system of equations:
x = -6/5
y = -1/5
Therefore, the solution to the given system of equations is x = -6/5 and y = -1/5.