How to Solve a System of Linear Equations | Methods and Outcomes Explained

system of linear equations

A system of linear equations refers to a set of two or more equations that involve multiple variables

A system of linear equations refers to a set of two or more equations that involve multiple variables. These equations are called linear because the highest power of any variable in the equation is 1 (e.g., x or y) and there are no terms involving products or powers of variables. The general form of a linear equation with two variables (x and y) is given as follows:

a₁x + b₁y = c₁
a₂x + b₂y = c₂
…
aₙx + bₙy = cₙ

Here, a₁, b₁, a₂, b₂, etc., are the coefficients, and c₁, c₂, etc., are the constants. The goal when solving a system of linear equations is to find the values of the variables (x and y, in this case) that satisfy all of the equations simultaneously. These values are known as a solution to the system of equations.

A system of linear equations can have multiple outcomes:

1. Unique Solution: If the system has exactly one solution (a unique value for x and y), then the equations intersect at a single point on the x-y coordinate plane.

2. No Solution: If the system has no solution, it means the equations are parallel and do not intersect. Graphically, this would appear as two lines that are parallel and never meet.

3. Infinite Solutions: If the system has infinitely many solutions, it means that the equations represent the same line or are completely overlapping. In this case, any point on the line represents a solution.

To solve a system of linear equations, various methods can be used, including substitution, elimination, or using matrices and row operations. These methods aim to simplify the system to a form where the solution(s) can be easily determined.

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