Kramer's Rule Lecture Notes

Overview

• Kramer's Rule is a method for solving systems of linear equations using determinants.
• Can be applied to systems with two variables and three variables.

Two-Variable System

• Equations:
  • Equation 1: a1x + b1y = c1
  • Equation 2: a2x + b2y = c2

Key Formulas

• Determinant (D):
  [ D = a1 \cdot b2 - a2 \cdot b1 ]

• Values of x and y:

  • [ x = \frac{D_x}{D} ]
  • [ y = \frac{D_y}{D} ]

Calculation Steps

1. Find D:

   • Use the coefficients from the left side:
     • [ D = \begin{vmatrix} a1 & b1 \\ a2 & b2 \end{vmatrix} = a1 \cdot b2 - a2 \cdot b1 ]

2. Find D_x:

   • Replace coefficients of x with c1 and c2:
     • [ D_x = \begin{vmatrix} c1 & b1 \\ c2 & b2 \end{vmatrix} = c1 \cdot b2 - c2 \cdot b1 ]

3. Find D_y:

   • Replace coefficients of y with c1 and c2:
     • [ D_y = \begin{vmatrix} a1 & c1 \\ a2 & c2 \end{vmatrix} = a1 \cdot c2 - a2 \cdot c1 ]

Example 1

• Given equations:

  1. 2x + 3y = 13
  2. 3x - 5y = -9

• Calculate D:

  • [ D = 2 \cdot (-5) - 3 \cdot 3 = -10 - 9 = -19 ]

• Calculate D_x:

  • [ D_x = \begin{vmatrix} 13 & 3 \\ -9 & -5 \end{vmatrix} = (13 \cdot -5) - (-9 \cdot 3) = -65 + 27 = -38 ]

• Calculate D_y:

  • [ D_y = \begin{vmatrix} 2 & 13 \\ 3 & -9 \end{vmatrix} = (2 \cdot -9) - (3 \cdot 13) = -18 - 39 = -57 ]

• Final Values:

  • [ x = \frac{D_x}{D} = \frac{-38}{-19} = 2 ]
  • [ y = \frac{D_y}{D} = \frac{-57}{-19} = 3 ]

Verification

• Plugging values back to original equations confirms the solution is correct.

Three-Variable System

• Equations:
  • General form:
    [ a1x + b1y + c1z = d1 ]
    [ a2x + b2y + c2z = d2 ]
    [ a3x + b3y + c3z = d3 ]

Key Formulas for 3 Variables

• Determinant (D):

  • Coefficient matrix: [ D = \begin{vmatrix} a1 & b1 & c1 \\ a2 & b2 & c2 \\ a3 & b3 & c3 \end{vmatrix} ]

• Values of x, y, and z:

  • [ x = \frac{D_x}{D}, y = \frac{D_y}{D}, z = \frac{D_z}{D} ]

Calculation Steps for 3 Variables

1. Calculate D using a 3x3 matrix determinant.
2. Calculate D_x, D_y, D_z similarly, replacing the appropriate columns with the constants from the equation.

Example 2

• Given equations:

  1. 3x - 2y + z = 2
  2. 4x + 3y - 2z = 4
  3. 5x - 3y + 3z = 8

• Final Values:

  • After calculating all determinants, the values are:
    • x = 1, y = 2, z = 3

Verification

• Check values in original equations to confirm accuracy.

Conclusion

• Kramer's Rule is a systematic way to solve systems of equations, especially useful in certain educational contexts.
• Requires careful calculation of determinants, highlighting the importance of accuracy in each step.