How to solve the system using the kramer method

The solution to the system of linear equations of the second order can be found by the Cramer method. This method is based on the calculation of the determinants of the matrices of a given system. By alternately calculating the main and auxiliary determinants, one can say in advance whether the system has a solution or if it is incompatible. When finding auxiliary determinants, the elements of the matrix are alternately replaced by its free terms. The solution to the system is found by simply dividing the determinants found.

Instruction manual

one

Write down the given system of equations. Make her matrix. In this case, the first coefficient of the first equation corresponds to the initial element of the first row of the matrix. The coefficients from the second equation make up the second row of the matrix. Free members are written in a separate column. Fill in this way all the rows and columns of the matrix.

2

Calculate the main determinant of the matrix. To do this, find the products of the elements located on the diagonals of the matrix. First, multiply all the elements of the first diagonal, located from the top left to the bottom right of the matrix element. Then calculate the second diagonal as well. Subtract the second from the first work. The result of the subtraction will be the main determinant of the system. If the main determinant is not equal to zero, then the system has a solution.

3

Then find the auxiliary determinants of the matrix. First calculate the first helper determinant. To do this, replace the first column of the matrix with the column of free terms of the system of equations being solved. After that, determine the determinant of the resulting matrix according to a similar algorithm, as described above.

4

Substitute the free terms for the elements of the second column of the original matrix. Calculate the second auxiliary determinant. The total number of these determinants should be equal to the number of unknown variables in the system of equations. If all obtained determinants of the system are equal to zero, it is believed that the system has many undetectable solutions. If only the main determinant is equal to zero, the system is incompatible and has no roots.

5

Find a solution to a system of linear equations. The first root is calculated as the quotient of dividing the first auxiliary determinant by the main determinant. Write down the expression and count its result. Calculate the second solution of the system in the same way, dividing the second auxiliary determinant by the main determinant. Record the results.