System of Linear Equations

This example shows you how to solve a system of linear equations in Excel. For example, we have the following system of linear equations:

5x	+	1y	+	8z	=	46
4x	-	2y			=	12
6x	+	7y	+	4z	=	50

In matrix notation, this can be written as AX = B

		5	1	8					x					46
with A =		4	-2	0		,	X =		y		,	B =		12
		6	7	4					z					50

If A^-1 (the inverse of A) exists, we can multiply both sides by A^-1 to obtain X = A^-1B. To solve this system of linear equations in Excel, execute the following steps.

1. Use the MINVERSE function to return the inverse matrix of A. First, select the range B6:D8. Next, insert the MINVERSE function shown below. Finish by pressing CTRL + SHIFT + ENTER.

Note: the formula bar indicates that the cells contain an array formula. Therefore, you cannot delete a single result. To delete the results, select the range B6:D8 and press Delete.

2. Use the MMULT function to return the product of matrix A^-1 and B. First, select the range G6:G8. Next, insert the MMULT function shown below. Finish by pressing CTRL + SHIFT + ENTER.

3. Put it all together. First, select the range G6:G8. Next, insert the formula shown below. Finish by pressing CTRL + SHIFT + ENTER.

4. If you have Excel 365 or Excel 2021, simply select cell G6, enter the same formula as above and press Enter. Bye bye curly braces.

Note: this dynamic array formula, entered into cell G6, fills multiple cells. Wow! This behavior in Excel 365/2021 is called spilling.