Question

Use Cramer's Rule to solve the system of linear equations for x and y.

kx+(1−k)y=3; (1−k)x+ky=1. Find x and y

Answer 1

To solve the system of linear equations using Cramer's Rule, we need to find the determinants of the coefficient matrix and the matrices obtained by replacing the columns with the constant terms. We can then use these determinants to calculate the values of x and y.

Step-by-step explanation:

Cramer's Rule is used to solve a system of linear equations. In this case, the given equations are:

kx + (1-k)y = 3
(1-k)x + ky
To use Cramer's Rule, we need to find the determinants of the coefficient matrix and the matrices obtained by replacing one column with the constant terms. Let's calculate these determinants.

First, calculate the determinant of the coefficient matrix:

| k 1-k | = k(1-k) - (1-k)(-k) = 2k - k^2

Next, calculate the determinant obtained by replacing the first column with the constant terms:

| 3 1-k | = 3(1-k) - (1)(-k) = 3 - 3k + k = 4 - 2k

Similarly, calculate the determinant obtained by replacing the second column with the constant terms:

| k 1 | = k(1) - (1-k)(k) = k - k + k^2 = k^2

Now, we can use Cramer's Rule to find the values of x and y:

x = | 4-2k |/| 2k - k^2 |
y = | k^2 |/| 2k - k^2 |

So, the values of x and y can be calculated using these formulas.