Problem 61 - Algebra and Trigonometry

Question

The process of solving a linear system in three variables using Cramer's Rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer's Rule to find the value for only one of the variables? Describe how this process might work, presenting a specific example with your description. Remember that your goal is still to find the value for each variable in the system.

Step-by-Step Solution

Answer

Yes, one can use Cramer's Rule for one of the variables like 'x' and then use back-substitution to find the rest of the variables like 'y' and 'z'. Validate the solutions by checking them against the original equations.

Step 1: Understand Cramer's Rule

Cramer's rule relies on determinants of matrices. For a system of linear equations, the rule can be used to find values of the variables by replacing the coefficients of each variable in the determinant with the constants of each equation and dividing it by the determinant of the coefficients.

Step 2: Apply Cramer's Rule to One Variable

Assuming we have a system of linear equations: \(Ax+By+Cz = D1\), \(Ex+Fy+Gz = D2\), \(Hx+Iy+Jz = D3\). First, find the determinant of the coefficient matrix, let's call it D. Then, to find 'x' using Cramer's rule, replace the coefficients of 'x' with constants in determinant and divide it by D.

Step 3: Solve for Other Variables

After you have the value of 'x', you can use back-substitution to find the other variables. Substitute the found 'x' value into equation 1 or 2 or 3 and solve for one of the other variables, say 'y'. Afterwards, substitute 'x' and 'y' into one of the equations to find 'z'.

Step 4: Validation

Verify your obtained solutions are correct by substituting 'x', 'y' and 'z' into all three original equations and ensuring that both sides of the equations are equal.