Write the system of equations as an augmented matrix. Then solve for x and y.
-4x-y=6
12x-2y=-8
Multiply first equation by 3.
-12x - 3y = 18
12x - 2y = -8
Add the two equations.
-5y = 10
Solve for y, then x.
To write the system of equations as an augmented matrix, we'll arrange the coefficients of the variables in a matrix form. The augmented matrix consists of the coefficients of the variables on the left side and the constants on the right side.
The system of equations is:
-4x - y = 6 ----> 1st equation
12x - 2y = -8 ----> 2nd equation
To form the augmented matrix, we'll combine the coefficients and the constant terms as follows:
| -4 -1 | 6 |
| 12 -2 | -8 |
Now, we can solve for x and y by performing row operations on the augmented matrix. The goal is to use elementary row operations to simplify the matrix to the point where it is in row-echelon or reduced row-echelon form.
Applying row operations, we can simplify the augmented matrix:
Multiply row 1 by 3:
| -4 -1 | 6 |
| 12 -2 | -8 |
↓
| -12 -3 | 18 |
| 12 -2 | -8 |
Add row 1 to row 2:
| -12 -3 | 18 |
| 12 -2 | -8 |
↓
| -12 -3 | 18 |
| 0 -5 | 10 |
Multiply row 2 by -1/5:
| -12 -3 | 18 |
| 0 -5 | 10 |
↓
| -12 -3 | 18 |
| 0 1 | -2 |
Multiply row 2 by 3 and add it to row 1:
| -12 -3 | 18 |
| 0 1 | -2 |
↓
| -12 0 | 12 |
| 0 1 | -2 |
Now, the augmented matrix is in row-echelon form.
To solve for x and y, we'll convert the augmented matrix back into the system of equations:
-12x = 12 ----> x = -1 (equation 1)
y = -2 (equation 2)
So, the solution to the system of equations is x = -1 and y = -2.