Cramer's Rule for 3×3 Linear Systems

Cramer's Rule for 3×3 Linear Systems

① Cramer's Rule for Larger Systems

Cramer's Rule is a method for solving linear systems where the number of equations equals the number of variables. This means the coefficient matrix will be square: 2×2, 3×3, 4×4, and so on.

Cramer's Rule is particularly valuable in computer systems because its algorithms are clear and systematic. We can program the solution method into clear computer algorithms or codes, allowing the computer to proceed step by step until it reaches the final solution.

② Setting Up the 3×3 System

Let's solve a 3×3 system using Cramer's Rule. We have three equations with three variables, so the coefficient matrix will be 3×3:

Notice that we have three equations and the number of equations equals the number of variables. This means we can solve this system using Cramer's Rule.

③ Extracting the Coefficient Matrix

From our system, we extract the coefficient matrix and calculate its determinant using the 3×3 determinant method we learned in previous lessons.

④ Applying Cramer's Rule for 3×3

Now we can set up the Cramer's Rule formulas for our three variables. Each variable equals a specific determinant divided by the main determinant:

Now we need to determine what matrices to use for , , and .

⑤ Finding D_x (Determinant for x)

For , we replace the entire x-column with the constants from the right side of our equations.

⑥ Finding D_y (Determinant for y)

For , we replace the y-column with the constants from the right side.

⑦ Finding D_z (Determinant for z)

Finally, for , we replace the z-column with the constants.

⑧ Final Solution and Verification

Let's summarize our complete solution and verify it works in all three original equations.