# How to Solve a System of Equations in Excel

To solve a system of equations in Excel, you can use the built-in matrix functions such as `MMULT()`

, `MINVERSE()`

, and `TRANSPOSE()`

. Here's a step-by-step guide on how to do this:

- Prepare your system of equations in the form Ax = B, where A is the matrix of coefficients, x is the column vector of unknowns, and B is the column vector of constants. For example, consider the following system of equations:

```
2x + 3y = 5
4x - y = 3
```

- In Excel, enter the coefficients and constants in separate cells. For instance, you can enter the coefficients in cells A1:B2 and the constants in cells C1:C2.

```
A1: 2 B1: 3 C1: 5
A2: 4 B2: -1 C2: 3
```

- Calculate the inverse of the coefficient matrix A. To do this, select a range of empty cells with the same size as the coefficient matrix (2x2 in our example). In this case, let's use cells E1:F2.
- With the range E1:F2 selected, type the following formula:

`=MINVERSE(A1:B2)`

- Press
`Ctrl`

+`Shift`

+`Enter`

to create an array formula. Excel calculates the inverse of the coefficient matrix and displays the result in cells E1:F2. - Now, you need to multiply the inverse of the coefficient matrix (in cells E1:F2) with the column vector of constants (in cells C1:C2). Select another range of empty cells with the same number of rows as the coefficient matrix and one column. In this case, let's use cells G1:G2.
- With the range G1:G2 selected, type the following formula:

`=MMULT(E1:F2, C1:C2)`

- Press
`Ctrl`

+`Shift`

+`Enter`

to create an array formula. Excel calculates the product of the matrix and the vector and displays the result in cells G1:G2. - The resulting column vector in cells G1:G2 now contains the solution to the system of equations. In our example, G1 contains the value of x, and G2 contains the value of y.

## Example

Here's an example using the system of equations mentioned earlier:

```
2x + 3y = 5
4x - y = 3
```

- Enter the coefficients and constants in cells A1:C2:

```
A1: 2 B1: 3 C1: 5
A2: 4 B2: -1 C2: 3
```

- Calculate the inverse of the coefficient matrix in cells E1:F2:

```
E1: 0.2 F1: 0.6
E2: 0.8 F2: -0.4
```

- Multiply the inverse of the coefficient matrix by the column vector of constants in cells G1:G2:

```
G1: 1
G2: 1
```

The solution to the system of equations is x = 1 and y = 1.

Did you find this useful?