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How to Solve Systems of Equations in Two Variables Using Determinants

Equations with two variables -- "X" and "Y" -- are given as "a1X + b1Y = c1" and "a2X + b2Y = c2," where the letters "a1," "a2," "b1," "b2," "c1" and "c2" denote the numeric equation coefficients. The solution of this system is a pair of values ("X" and "Y") that simultaneously satisfy both equations. In mathematics, Cramer's rules allow you to easily solve such equations. The procedure is based on computing determinants for three equation coefficient matrices.

Things You'll Need

Calculator

Instructions

- 1
  Write down the system of the equations with two variables; for example:
  
  2X - 5Y = 10
  
  3X + 8Y =25
  
  The equation coefficients are: a1=2, b1=-5, c1=10, a2=3, b2=8 and c2=25.
- 2
  Calculate the determinant of the first matrix using the expression: a1 x b2 - a2 x b1. In this example, the determinant is: 2 x 8 - 3 x (-5) = 31.
- 3
  Calculate the second determinant using the expression: c1 x b2 - c2 x b1. In this example, the determinant is: 10 x 8 - 25 x (-5) = 205.
- 4
  Calculate the third determinant using the expression: a1 x c2 - a2 x c1. In this example, the determinant is: 2 x 25 - 3 x 10 = 20.
- 5
  Divide the second determinant by the first one to calculate the value of the variable "X." In this example: "X" is 205/31 = 6.613.
- 6
  Divide the third determinant by the first one to calculate the value of the variable "Y." In this example: "Y" is 20/31 = 0.645.

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