Answer and Step-by-step explanation:
To solve the system of equations using the matrix method, we can represent the equations in matrix form.
First, let's write the system of equations:
2x - 3y = -17 (Equation 1)
5x + 6y = -2 (Equation 2)
Next, we can represent the coefficients and constants of the equations in matrix form:
| 2 -3 | | x | = | -17 |
| 5 6 | | y | | -2 |
Now, let's write the augmented matrix, which combines the coefficient matrix and the constant matrix:
| 2 -3 -17 |
| 5 6 -2 |
To solve the system, we will perform row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form.
Step 1: Divide Row 1 by 2 to make the leading coefficient of the first row 1:
| 1 -3/2 -17/2 |
| 5 6 -2 |
Step 2: Replace Row 2 with Row 2 - 5 * Row 1:
| 1 -3/2 -17/2 |
| 0 21/2 33/2 |
Step 3: Divide Row 2 by 21/2 to make the leading coefficient of the second row 1:
| 1 -3/2 -17/2 |
| 0 1 3 |
Step 4: Replace Row 1 with Row 1 + (3/2) * Row 2:
| 1 0 -2 |
| 0 1 3 |
The augmented matrix is now in reduced row-echelon form. The leftmost column represents the coefficients of x and y, respectively, while the rightmost column represents the constant terms.
From the reduced row-echelon form, we can read the solution of the system:
x = -2
y = 3
Therefore, the solution to the system of equations is x = -2 and y = 3.