# How to Solve a System of Equations in R (3 Examples)

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.