System of linear equations

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

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, making the system true.

Linear systems are a cornerstone of linear algebra, which is used in nearly every area of modern science and technology. They help solve problems in engineering, physics, chemistry, computer science, and economics. Even complicated systems that are not linear can often be simplified by approximating them with linear systems, making them easier to study and solve.

Elementary examples

A system of linear equations is a set of equations where each one is straight and uses the same variables. For example, the simple equation 2x = 4 has just one solution: x = 2.

More interesting systems have two or more equations with two variables. For instance:

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 make all the equations true at the same time. This method can also be used for systems with more variables.

Main article: elementary algebra

General form

A system of linear equations is a set of equations where we try to find values for variables that make all equations true at the same time. 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 satisfy 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 involve real numbers, complex numbers, integers, or even more abstract mathematical objects.

Solution set

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

Linear systems can have three different outcomes: they can have infinitely many solutions, one unique solution, or no solution at all. For two variables, each equation represents a line, and the solution is where the lines meet — either at a single 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, like multiplying everything in an equation by two, they don't 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, like "3x + 2y equals both 6 and 12," 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 can be described by specific values for each variable, like (x = 3), (y = -2), and (z = 6). Sometimes, there are infinitely many solutions, and some variables can take any value while others depend on these free variables.

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

Another common method is row reduction, also known as Gaussian elimination. This involves representing the system as a matrix and using specific operations to simplify it until the solution becomes clear. This method systematically reduces 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 because we are looking for values of the variables that add up to zero.

Every homogeneous system has at least one solution: the zero solution, where every variable is set to zero. Sometimes, there are many more solutions. If there are extra solutions, they follow special rules: adding two solutions together gives another solution, and multiplying a solution by a number also gives a solution. These solutions form a shape called a linear subspace.