To solve a system of equations in R, we can use the built-in **solve()** function.

The following examples show how to use this functions to solve several different systems of equations in R.

**Example 1: Solve System of Equations with Two Variables**

Suppose we have the following system of equations and we’d like to solve for the values of x and y:

5x + 4y = 35

2x + 6y = 36

The following code shows how to use the **solve()** function in R to solve for the values of x and y:

#define left-hand side of equations left_matrix <- matrix(c(5, 2, 4, 6), nrow=2) left_matrix [,1] [,2] [1,] 5 4 [2,] 2 6 #define right-hand side of equations right_matrix <- matrix(c(35, 36), nrow=2) right_matrix [,1] [1,] 35 [2,] 36 #solve for x and y solve(left_matrix, right_matrix) [,1] [1,] 3 [2,] 5

This tells us that the value for x is **3** and the value for y is **5**.

**Example 2: Solve System of Equations with Three Variables**

Suppose we have the following system of equations and we’d like to solve for the values of x, y, and z:

4x + 2y + 1z = 34

3x + 5y – 2z = 41

2x + 2y + 4z = 30

The following code shows how to use the **solve()** function in R to solve for the values of x, y, and z:

#define left-hand side of equations left_matrix <- matrix(c(4, 3, 2, 2, 5, 2, 1, -2, 4), nrow=3) left_matrix [,1] [,2] [,3] [1,] 4 2 1 [2,] 3 5 -2 [3,] 2 2 4 #define right-hand side of equations right_matrix <- matrix(c(34, 41, 30), nrow=3) right_matrix [,1] [1,] 34 [2,] 41 [3,] 30 #solve for x, y, and z solve(left_matrix, right_matrix) [,1] [1,] 5 [2,] 6 [3,] 2

This tells us that the value for x is **5**, the value for y is **6**, and the value for z is **2**.

**Example 3: Solve System of Equations with Four Variables**

Suppose we have the following system of equations and we’d like to solve for the values of w, x, y, and z:

6w + 2x + 2y + 1z = 37

2w + 1x + 1y + 0z = 14

3w + 2x + 2y + 4z = 28

2w + 0x + 5y + 5z = 28

The following code shows how to use the **solve()** function in R to solve for the values of w, x, y, and z:

#define left-hand side of equations left_matrix <- matrix(c(6, 2, 3, 2, 2, 1, 2, 0, 2, 1, 2, 5, 1, 0, 4, 5), nrow=4) left_matrix [,1] [,2] [,3] [,4] [1,] 6 2 2 1 [2,] 2 1 1 0 [3,] 3 2 2 4 [4,] 2 0 5 5 #define right-hand side of equations right_matrix <- matrix(c(37, 14, 28, 28), nrow=4) right_matrix [,1] [1,] 37 [2,] 14 [3,] 28 [4,] 28 #solve for w, x, y and z solve(left_matrix, right_matrix) [,1] [1,] 4 [2,] 3 [3,] 3 [4,] 1

This tells us that the value for w is **4**, x is **3**, y is **3**, and z is **1**.

**Additional Resources**

The following tutorials explain how to perform other common operations in R:

How to Calculate Five Number Summary in R

How to Create Summary Tables in R

How to Calculate Z-Scores in R