In mathematical terms, the system of equation is set of two or more equations having the same set of unknown variables like x, y, z where we need to find the values of unknown variables to solve these equations. To solve the system of equations, we can utilize functions and the equation solver tool.

*Figure 1. How to Solve the System of Equations*

**Using Excel Functions**

We can solve the system of equations by using MINVERSE and MMULT mathematical functions. Suppose we have three equations in our system of equations in our example. To solve equations using functions we need to set up the data as follows;

* Figure 2. Setting Up Data of System of Equations*

In matrix notation form we can write it as;

**AX= B**

And to find the value of unknown variables **X** we need to find the inverse of matrix **A** (**A****-1**) and multiply it with matrix **B**, such as;

**X= A****-1****B**

To find the inverse of matrix A (**A****-1**) we need to use the MINVERSE function as follows;

- Select the range of cells as per the matrix A layout such as
**C10: E12**

* Figure 3. Selecting Range of Cells*

- Go to the formula bar and insert the
**MINVERSE**formulas as

and press**=MINVERSE(C6: E8)****Ctrl + Shift + Enter**instead of pressing Enter key to make it an array formula to return an array of value as the inverse of matrix A.

* Figure 4. Finding Inverse of Matrix A*

Now we need to find the values of unknown variables in **matrix X** (x, y, z) in column H to solve the system of equations using Excel MMULT function as follows;

- Select the range of cells
**H6: H8**of matrix X

* Figure 5. Selecting Range of Cells of Matrix X*

- In the formula bar, insert the
**MMULT**formula as

to return the product of two matrix A-1 (inverse of Matrix A) and B by using its arrays and press Ctrl + Shift + Enter to make it an array formula. This formula returns the values of unknown variables and solves the system of equations.**=MMULT(C10: E12, K6: K8)**

* Figure 6. Solve System of Equations*

**Using Excel Solver Add-in**

Another approach to solve the system of equations is Excel Solver Add-in. We need to install this add-in from available add-ins from **Options > Add-ins**. For Excel Solver we need to set up the data of the system of equations as follows;

- For matrix A we need to enter the formula for each of the equations in
**column C**as shown. - Insert the values of matrix B in
**column D**. - Leave the cells of matrix X blank in
**column E**to return the values of unknown variables**x, y, z**.

* Figure 7. Setting Up Data of Equations for Solver*

- Go to
**Data**tab, from**Analysis**group select the**Solver**to launch.

* Figure 8. Solver Parameters Window*

- In the
**Set Objective**box insert the cell reference**C7**, the first cell of matrix A where we have set the formula for the first equation of the system. - Select the objective as
**Max**. - Insert the range of cell references
**E7: E9**of**matrix****X**in the**By Changing Variable Cells**box.

* Figure 9. Setting Up Solver Parameters*

- In the
**Subject to Constraints**box, we need to add the constraints for each of the equations by clicking on**Add**button. - In the
**Add Constraints**window, we need to enter the constraints for all equations one by one by pressing**Add**button. Press the**OK**button when we are done with adding all the constraints.

* Figure 10. Adding Constraints in Solver*

- After adding all the constraints, we need to
**unselect**the**Make Unconstrained Variables Non-negative**checkbox below the constraints box. - Press the
**Solve**button

* Figure 11. Solving Equations*

- On
**Solver Results**window, select**Keep Solver Solutions**and press**OK**button

* Figure 12. Solver Results*

- Solver returns the values of
**unknown variables**for**matrix X**in column E

* Figure 13. Equation Solver*

**Instant Connection to an Expert through our Excelchat Service**

Most of the time, the problem you will need to solve will be more complex than a simple application of a formula or function. If you want to save hours of research and frustration, try our live Excelchat service! Our Excel Experts are available 24/7 to answer any Excel question you may have. We guarantee a connection within 30 seconds and a customized solution within 20 minutes.

## Leave a Comment