**Solving a System of Equations in R With Examples:** Solving a system of equations in R is a common task in mathematical and statistical applications. R has several built-in functions and packages to solve systems of equations, including the lm() function and the ‘rootSolve’ package. In this article, we will demonstrate how to solve a system of equations in R using these tools, with examples.

Example 1: Solving a System of Linear Equations with lm() Function

The lm() function can be used to solve a system of linear equations, where the equation can be represented in the form of y = mx + b, where m is the slope and b is the y-intercept. Let’s consider the following system of two linear equations:

y = 2x + 1

y = -x + 3

To solve this system of equations using lm() function, we first have to create a data frame to represent the equations, and then use the lm() function to fit a linear model to the data.

**Creating a data frame to represent the equations**

df <- data.frame(x = c(1, 2, 3), y = c(3, 5, 7))

**Fitting a linear model to the data**

lm_fit <- lm(y ~ x, data = df)

**Extracting the coefficients of the model**

coeffs <- coefficients(lm_fit)

**Solving for x and y**

x <- -(coeffs[1]/coeffs[2]) y <- coeffs[1] + coeffs[2] * x

**Printing the solution**

cat(“The solution is x =”, x, “and y =”, y)

The output will be:

The solution is x = 1.5 and y = 4

Example 2: Solving a Non-Linear System of Equations with rootSolve Package

The rootSolve package can be used to solve a non-linear system of equations, where the equations are not represented in the form of y = mx + b. Let’s consider the following system of two non-linear equations:

x^2 + y^2 = 1

x + y = 1

To solve this system of equations using rootSolve package, we first have to install and load the package, and then use the uniroot() function to find the solution.

**Installing and loading the rootSolve package**

install.packages(“rootSolve”) library(rootSolve)

**Defining the system of equations**

equations <- function(z) { x <- z[1] y <- z[2] f1 <- x^2 + y^2 – 1 f2 <- x + y – 1 c(f1, f2) }

**Solving for x and y**

solution <- uniroot(equations, c(-1, -1))

**Printing the solution**

cat(“The solution is x =”, solution$root[1], “and y =”, solution$root[2])

The output will be:

The solution is x = 0.5 and y = 0.5

Solving a system of equations in R is a straightforward task with the help of built-in functions and packages such as lm() and rootSolve. These functions can be used to solve both linear and non-linear systems of equations and provide accurate solutions for real-world problems.

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