System of linear equations

In mathematics, a system of linear equations (or linear system) is a group of two or more linear equations that use the same variables. These equations are like puzzles. We need to find values for the variables that make all the equations true together. For example, a system might ask us to find numbers for x, y, and z that work in three different equations at the same time.

A solution to a linear system is a set of values for the variables that works for every equation in the system. In the example given, the values x = 1, y = −2, and z = −2 solve all three equations. This makes the system true.

Linear systems are important in linear algebra. They are used in many areas, such as engineering, physics, chemistry, computer science, and economics. Even complicated systems that are not linear can often be made simpler by using linear systems. This makes them easier to study and solve.

Elementary examples

A system of linear equations is a set of straight equations that use the same variables. For example, the equation 2x = 4 has one solution: x = 2.

Systems can have two or more equations with two variables. For example:

2x + 3y = 6
4x + 9y = 15

To solve this, we can find the value of x from the first equation and then use it in the second equation. This helps us find the values of both x and y that work for all the equations at the same time. We can use this method for systems with more variables too.

Main article: elementary algebra

General form

A system of linear equations is a set of equations where we look for values for variables that make all equations true together. For example, we might have three equations with three unknowns, like x, y, and z. Solving the system means finding values for these variables that work for every equation.

We can write a general system of m linear equations with n unknowns and coefficients in a compact form. This helps us see patterns and use tools from vector spaces and matrices to solve the equations. The system can use real numbers, complex numbers, integers, or other mathematical objects.

Solution set

A solution of a linear system is when we find numbers for the variables that make all the equations true together. The set of all possible solutions is called the solution set.

Linear systems can have three different results: they can have many solutions, one special solution, or no solution at all. For two variables, each equation is a line, and the solution is where the lines meet — either at one point, along a line, or not at all if they never cross.

Properties

A system of linear equations has special properties that help us understand its solutions.

Independence means that none of the equations can be made from the others by simple math steps. If equations are independent, each one gives new information. If they are not independent, they do not add new information.

Consistency describes whether a system has a solution. If there is no way to make the equations all true at once, the system is inconsistent. For example, two equations that say the same thing but with different answers cannot both be true, so they are inconsistent.

Solving a linear system

There are several ways to solve a system of linear equations. When the solution is unique, it gives special numbers for each variable, like (x = 3), (y = -2), and (z = 6). Sometimes, there are many solutions, and some variables can be any number while others change based on these.

One simple method is to remove variables step by step. You solve one equation for a variable, then use that in the other equations, repeating until you have just one equation left to solve. For example, solving a system of three equations can give a single solution like ((x, y, z) = (-15, 8, 2)).

Another common method is row reduction, also known as Gaussian elimination. This uses a matrix to represent the system and simplifies it with special steps until the solution is easy to see. This method organizes the equations to find the values of the variables.

Homogeneous systems

See also: Homogeneous differential equation

A homogeneous system of linear equations is one where all the numbers on the right side of the equations are zero. For example, instead of saying 3x + 2y – z = 1, we might have 3x + 2y – z = 0. This makes the equations simpler.

Every homogeneous system has at least one solution: the zero solution, where every variable is set to zero. Sometimes, there are more solutions. These solutions follow special rules. They form a shape called a linear subspace.