12. Linear systems

Linear systems are a set of equations that are solved simultaneously. In the context of Mathematics for the ENEM exam, it is important to understand how to solve these systems to deal with practical and theoretical problems.

A linear system can be represented as Ax = b, where A is the matrix of coefficients, x is the vector of variables, and b is the vector of constants. The objective is to find the vector x that satisfies all the equations in the system. There are several methods for solving linear systems, including substitution, elimination, Cramer's rule, and matrix inverse.

The replacement method is usually the simplest. You start by solving one of the equations for a variable and then plug that expression into the other equation. This results in a new equation with only one variable, which can be solved easily. The solution for the other variable is then found by substituting the first solution into the original equation.

The elimination method, on the other hand, involves adding or subtracting the equations to eliminate one of the variables. This also results in a new equation with only one variable. The solution for the other variable is then found by substituting the first solution into the original equation.

Cramer's rule is a more advanced method that uses determinants of matrices. For a system of n equations with n variables, the determinant of the coefficient matrix is ​​computed. Then, for each variable, the corresponding column in the matrix of coefficients is replaced by the vector of constants and the determinant of this new matrix is ​​calculated. The solution for each variable is then the quotient of the determinant of the modified matrix by the determinant of the original matrix.

Matrix inverse is an even more advanced method that can only be used when the coefficient matrix is ​​invertible. In this case, the inverse matrix is ​​multiplied by the vector of constants to find the solution for the vector of variables.

It is important to note that not all linear systems have a solution. If the determinant of the coefficient matrix is ​​zero, then the system is said to be inconsistent and has no solution. If the system has more variables than equations, then it has infinitely many solutions.

On the ENEM test, linear systems can appear in a variety of contexts, from geometry problems to economics issues. Therefore, it is important to be comfortable with all resolution methods and understand when to use each one.

Also, it is useful to know how to interpret the solution of a linear system. For example, in a geometry problem, solutions might represent points of intersection of lines or planes. In an economics problem, solutions may represent quantities of goods or services.

In summary, linear systems are an important tool in mathematics that can be used to solve a wide range of problems. Mastering this skill can be very useful for the ENEM exam and for other practical and theoretical applications of mathematics.

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