Use a software program -8x1 7x2 12x1 + 3x2 15x₁ - 9x2 + + (X1, X2, X3) = 1 or a graphing utility with matrix capabilities and Cramer's Rule to solve (if possible) the 10x3 = -142 5x3 = -21 2x3 = 199
Given:
The system of linear equations are:
Cramer's Rule:
Cramer's Rule is a method for solving a system of linear equations by using determinants. Specifically, it provides a formula for finding the solution to a system of equations in terms of the determinants of matrices formed from the coefficients of the equations.
Assume that we have a system of n linear equations in n variables:
We can write this system in matrix form as Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants.
Cramer's Rule states that the solution for the variable can be found by taking the determinant of the matrix obtained by replacing the i-th column of A with the column vector b, and dividing by the determinant of A. That is,
,
where A_i is the matrix obtained by replacing the i-th column of A with b.
Cramer's Rule has some limitations and is not always the most efficient method for solving a system of linear equations, but it can be useful in some situations, especially in small systems.
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