## Saturday, February 24, 2018

### Cramer’s Rule Two Equations and Two Variables in Excel

Cramer’s Rule Two Equations and Two Variables in Excel

Question:   Consider the system of equations

2x+ 5y=10

7x-6y=-5

Discuss how this system is solved using Cramer’s Rule

Find the (x, y) pair solution for this two-equation two variable system.

Solve using Cramer’s rule with the MDETERM function in Excel.

Create a spreadsheet that can be easily modified for a system of two equations and two variables.

Discussion of Cramer’s Rule:

Cramer’s rule states that x and y are the ratio of two determinants.

The denominator of both x and y from Cramer’s rule is the determinant of the matrix composed of the x and y coefficients.   We denote this determinant D.

 Matrix for denominator of X and Y 2 5 7 -6

The numerator of x from Cramer’s rule is the determinant of the matrix where the constant terms are inserted in the left column and the y coefficients are in the right column.  We denote this matrix Dx
 Matrix for Numerator of X 10 5 -5 -6

The numerator of y from Cramer’s rule is the determinant of the matrix where the constant terms are in the right column and the coefficients of the x variables are in the left column.   We denote this matrix Dy is

 Matrix for Numerator of Y 2 10 7 -5

Discussion of Spreadsheet Solution for this Problem:

The Excel function MDETERM(array) was used to calculate D, Dx and Dy.   See chart below
 Matrix Array Input Solution from MDETERM Function D 2 5 -47 7 -6 Dx 10 5 -35 -5 -6 Dy 2 10 -80 7 -5 X (Dx/D) 0.745 Y (Dy/D) 1.702

The spreadsheet has an input section for coefficients of x, y and the constant term.  Change the values of the spreadsheet to solve any system with two equations and two variables.