The Cramer's rule is a method used to solve a system of linear equations. Given a system of two linear equations in two variables:
a1x + b1y = c1
a2x + b2y = c2
where a1, b1, c1, a2, b2, and c2 are constants, and x and y are the variables we want to solve for, we can use Cramer's rule to find the values of x and y.
Cramer's rule states that the solution for x is given by:
x = (det(Ax)) / (det(A))
where Ax is the matrix obtained by replacing the x column in the coefficient matrix with the constants c1 and c2, and det(Ax) is the determinant of Ax.
Similarly, the solution for y is given by:
y = (det(Ay)) / (det(A))
where Ay is the matrix obtained by replacing the y column in the coefficient matrix with the constants c1 and c2, and det(Ay) is the determinant of Ay.
Now, let's apply Cramer's rule to the system of linear equations:
x - y = -3
We can rewrite this equation as:
x + (-1)y = -3
So, the coefficient matrix A and the constants matrix B are:
A = | 1 -1 |
| 0 0 |
B = | -3 |
| 0 |
The determinant of A is:
det(A) = (1 * 0) - (0 * -1) = 0
Since the determinant is zero, Cramer's rule cannot be applied to this system of equations. This means that either the system has no solution or it has infinitely many solutions. In this case, we can see that the equation x - y = -3 represents a line in the xy-plane with slope 1 and y-intercept -3. This line does not intersect with the y-axis, which means that there is no solution to the system of equations.
Therefore, the system of equations x - y = -3 has no solution.