Linear equation

In mathematics, a linear equation is a special kind of equation that helps us understand relationships between numbers. It can be written in a simple form where we add up multiplied numbers and variables, and set the total equal to zero. The variables are the unknown values we want to find, and the numbers multiplying them are called coefficients. These coefficients are usually real numbers, meaning they are numbers we use in everyday counting and measuring.

When we have just one variable, a linear equation has one solution, which is the value that makes the equation true. If we work with two variables, each solution can be thought of as a point on a graph. All the solutions to a linear equation with two variables form a straight line on this graph. This is why these equations are called "linear"—because they describe straight lines. In higher dimensions, the solutions form what is called a hyperplane.

Linear equations are very important because they appear often in many areas, such as physics and engineering. Even when problems are complicated and not linear, we can often approximate them using linear equations to make them easier to solve. This makes linear equations a fundamental tool in both mathematics and its many applications. For situations with several linear equations at the same time, we use a system of linear equations, which is discussed in a separate article system of linear equations.

One variable

A linear equation with just one variable, like x, can be written as ax + b = 0, where a and b are numbers, and a is not zero. To solve this equation and find x, you can rearrange it to x = -b/a. This means you divide -b by a to get the value of x.

Two variables

A linear equation in two variables, like x and y, can be written as a x + b y + c = 0, where a and b are not both zero. If a and b are real numbers, this equation has many solutions.

Main article: Linear function (calculus)

When b is not zero, the equation can be solved for y in terms of x, giving y = −a/b_ x − c/b_. This shows that for each value of x, there is exactly one value of y, forming a straight line when plotted on a graph. The slope of this line is −a/b_, and it crosses the y-axis at −c/b_. Such equations are called linear functions.

More than two variables

A linear equation with more than two variables can always be written in a special form. This form looks like this: a₁x₁ + a₂x₂ + ... + aₙxₙ + b = 0. Here, the letters x₁, x₂, ..., xₙ are the variables, and a₁, a₂, ..., aₙ and b are numbers called coefficients.

When we have three variables, we usually use x, y, and z instead of x₁, x₂, and x₃. To solve such an equation means finding values for the variables that make the equation true. If all the coefficients next to the variables are zero, the equation either has no solution or every possible set of values is a solution, depending on the constant term b.

The solutions to a linear equation with n variables can be thought of as points that lie on a flat surface in n-dimensional space. For three variables, this flat surface is a plane. If at least one coefficient next to a variable is not zero, we can solve the equation for that variable, expressing it in terms of the other variables.