Use Cramer's rule to solve the following system of equations. If D=0, use another method to solve the system.
x+y=11
x-y=5
x =
|11 1|
|5 -1|
------
|1 1|
|1 -1|
= -16/-2 = 8
y = 11-x = 3
The | | /| | symbols represent the ratios of two 2x2 determinants
To solve the system of equations using Cramer's rule, we start by writing the equations in matrix form:
| 1 1 | | x | | 11 |
| 1 -1 | * | y | = | 5 |
To find the values of x and y, we need to calculate three determinants: D, Dx, and Dy.
1. Calculate the determinant D of the coefficient matrix:
D = | 1 1 |
| 1 -1 |
Using the formula for a 2x2 determinant, D = (1 * -1) - (1 * 1) = -2.
2. Calculate the determinant Dx of the matrix formed by replacing the x-column with the constant column:
Dx = | 11 1 |
| 5 -1 |
Using the formula for a 2x2 determinant, Dx = (11 * -1) - (1 * 5) = -16.
3. Calculate the determinant Dy of the matrix formed by replacing the y-column with the constant column:
Dy = | 1 11 |
| 1 5 |
Using the formula for a 2x2 determinant, Dy = (1 * 5) - (1 * 11) = -6.
Now, we can find the values of x and y using Cramer's rule:
x = Dx / D = -16 / -2 = 8
y = Dy / D = -6 / -2 = 3
Therefore, the solution to the system of equations is x = 8 and y = 3.
However, if the determinant D is equal to 0, that means Cramer's rule cannot be applied, and we need to use another method to solve the system. In this case, we can add the two equations together to eliminate the y variable:
(x + y) + (x - y) = 11 + 5
2x = 16
x = 8
Substituting the value of x into one of the equations:
8 + y = 11
y = 11 - 8
y = 3
So the solution to the system of equations is x = 8 and y = 3.