Cramer’s Rule Two
Equations and Two Variables in Excel
Question: Consider the system of equations
2x+ 5y=10
7x6y=5
Discuss how this system is solved using Cramer’s Rule
Find the (x, y) pair solution for this twoequation 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.

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 D_{x}.

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 D_{y} 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, D_{x}
and D_{y}. See chart below
Matrix

Array Input

Solution from MDETERM Function


D

2

5

47

7

6


D_{x}

10

5

35

5

6


D_{y}

2

10

80

7

5


X (D_{x}/D)

0.745


Y (D_{y}/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.
Go to this web site to download the spreadsheet described
here.
Remember there is no solution when the determinant of the
coefficient matrix is 0.
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