| Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables |
First let Take note that the right hand values of the system are These values are important as they will be used to replace the columns of the matrix A. Now let's calculate the the determinant of the matrix A to get Notation note: --------------------------------------------------------- Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix Now compute the determinant of To find the first solution, simply divide the determinant of So the first solution is --------------------------------------------------------- We'll follow the same basic idea to find the other two solutions. Let's reset by letting Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix Now compute the determinant of To find the second solution, divide the determinant of So the second solution is --------------------------------------------------------- Let's reset again by letting Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix Now compute the determinant of To find the third solution, divide the determinant of So the third solution is ==================================================================================== Final Answer: So the three solutions are Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant. |