Use Cramer’s Rule to solve (if possible) the system of equations. { -0.4 x+0.8 y =1.6 0.2 x

step 1 Given the system negative 0 .4x plus 0 .8 y equals 1 .6 and 0 .2x plus 0 .3y equals 2 .2, we wan

Use Cramer’s Rule to solve (if possible) the system of...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables. The goal is to find values for the variables that satisfy all the equations simultaneously. These systems can be represented in various forms and are foundational in linear algebra, with applications in fields ranging from engineering to economics.

Matrix Representation

Matrix representation converts a system of linear equations into a compact, structured format using matrices. The coefficients of the variables form the coefficient matrix, while the variables and constants are organized into vectors. This representation facilitates the use of various algebraic techniques and computational methods for solving the system.

Determinant

A determinant is a scalar value computed from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible. In the context of a system of linear equations, the determinant of the coefficient matrix determines whether a unique solution exists; if the determinant is nonzero, the system has a unique solution.

Cramer’s Rule

Cramer's Rule is a method for solving a system of linear equations using determinants. It states that each variable in the system can be solved by taking the determinant of a matrix formed by replacing the corresponding column of the coefficient matrix with the constants vector, divided by the determinant of the original coefficient matrix. This rule provides a direct computational formula for the solutions when the determinant of the coefficient matrix is nonzero.

Solution Existence

The existence of a solution to a system of linear equations is determined by the determinant of the coefficient matrix. If the determinant is nonzero, a unique solution exists. If it is zero, the system may either have no solutions or infinitely many solutions, depending on the relationship between the equations. This concept is crucial for determining which methods for solving the system are applicable.

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