Final answer:
Cramer's rule is a method used to solve a system of linear equations by using determinants. It involves finding determinants of matrices to determine the values of the variables in the system. The steps include writing the system of equations in matrix form, finding the main determinant, and calculating the determinants of the variables using column replacements.
Step-by-step explanation:
Cramer's rule is a method used to solve a system of linear equations by using determinants. To use Cramer's rule, you need to have the same number of equations as variables. Here are the steps to solve a system of equations using Cramer's rule:
- Write the system of equations in matrix form, where the coefficients of the variables are the elements of the matrix.
- Find the determinant of the coefficient matrix, also known as the main determinant.
- Replace one column of the coefficient matrix with the constants from the right side of the equations, while keeping the rest of the columns the same. Find the determinant of this new matrix, known as the determinant of the variable you are solving for.
- Repeat step 3 for each variable by replacing a different column each time.
- The solution for each variable is found by dividing the determinant of the variable by the main determinant.
For example, let's say we have the system of equations:
2x + 3y = 8
4x - 2y = 2
The coefficient matrix is:
[2 3]
[4 -2]
The main determinant is
det([2 3] [4 -2]) = 2(-2) - 3(4) = -14
To find the determinant of the variable x, we replace the first column with the constants:
[8 3]
[2 -2]
The determinant of this matrix is:
det([8 3] [2 -2]) = 8(-2) - 3(2) = -22
The value of x is -22 / -14 = 11 / 7
To find the determinant of the variable y, we replace the second column with the constants:
[2 8]
[4 2]
The determinant of this matrix is:
det([2 8] [4 2]) = 2(2) - 4(8) = -28
The value of y is -28 / -14 = 2