Solve systems of linear equations in Excel without using VBA

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Suppose you have the following system of linear equations and want to solve for x, y, and z:

  • 2x + y – z = -1
  • -4x + 11y = 18
  • x – y + 2z = 9

Is there an easy way to do this with Excel using worksheet functions?


This can be done by using a little linear algebra and Excel’s matrix functions.

First, create an augmented matrix that represents your equations on a worksheet.

Since you want to solve for three variables, select 3 cells in a column where you want your results to be returned. Without changing the selection, type the following formula:


Press Ctrl+Shift+Enter to enter the formula in the selected array. And that will return the solution to your system of equations.

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This will work for larger systems of equations with more variables as well. This formula works based on the observation that the left-hand side of a system of linear equations can be represented as the product of an n by n matrix, A, and an n by 1 matrix of variables, x, and the right-hand side can be represented as an n by 1 matrix, B. To solve this matrix equation, both sides must be multiplied by the inverse of A.

Ax = B
inverse(A)Ax = inverse(A)B
x = inverse(A)B

Thus, the solutions are just the matrix product of the inverse of the square matrix and the matrix of the constants on the right-hand side of the equations.

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