Use Cramer's rule to solve the system.
-2x - 2y = -16
2x + 3y = 23
Cramer’s Rule for Two Equations in Two Variables:
(Broken Link Removed)
To use Cramer's rule to solve this system of equations, we need to find the determinants of the coefficient matrix and the determinant of each modified matrix.
Let's start by writing down the given system of equations:
-2x - 2y = -16
2x + 3y = 23
Now let's find the determinants:
First, we need to find the determinant of the coefficient matrix (denoted as D):
D = | -2 -2 |
| 2 3 |
The determinant D can be calculated as D = (-2 * 3) - (-2 * 2) = -6 - (-4) = -2.
Next, let's find the determinant Dx, which is obtained by replacing the coefficients of x with the constants of the system:
Dx = | -16 -2 |
| 23 3 |
It is calculated as Dx = (-16 * 3) - (-2 * 23) = -48 - (-46) = -48 + 46 = -2.
Similarly, let's find the determinant Dy, which is obtained by replacing the coefficients of y with the constants of the system:
Dy = | -2 -16 |
| 2 23 |
It is calculated as Dy = (-2 * 23) - (-16 * 2) = -46 - (-32) = -46 + 32 = -14.
Now, we can find the solutions for x and y using Cramer's rule:
x = Dx / D = (-2) / (-2) = 1
y = Dy / D = (-14) / (-2) = 7
Therefore, the solution to the system of equations is x = 1 and y = 7.