Systems of linear equations are mathematical equations that involve two or more linear equations with the same variables. These equations are used to represent a relationship between different variables and can be solved to find the values of the variables that satisfy all the equations.
A linear equation is a mathematical expression that describes a straight line on a graph. It has only one variable and can be solved to find the value of that variable. A system of linear equations, on the other hand, involves two or more linear equations with the same variables and can be solved to find the values of all the variables that satisfy all the equations.
The solution to a system of linear equations is the set of values for the variables that make all the equations in the system true. In other words, it is the point or points of intersection between the lines represented by the equations. A system of linear equations can have one unique solution, no solution, or infinitely many solutions.
There are several methods for solving a system of linear equations, including substitution, elimination, and graphing. In substitution, one equation is solved for one variable and then substituted into the other equation. In elimination, the equations are manipulated to eliminate one of the variables. In graphing, the equations are plotted on a graph and the point of intersection is found. Other methods, such as matrices and determinants, can also be used.
Systems of linear equations have many real-life applications, such as in economics, engineering, and physics. They can be used to model and solve problems involving multiple variables, such as budgeting, production planning, and electrical circuits. They are also used in data analysis and prediction, such as in regression analysis and linear programming.