In Problems 41-60, use the inverses found in Problems 31-40...

In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations.
$\left\{\begin{array}{r}x+y+z=2 \\ 3 x+2 y-z=\frac{7}{3} \\ 3 x+y+2 z=\frac{10}{3}\end{array}\right.$

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Key Concepts

System of Linear Equations

This concept involves multiple equations that are solved simultaneously for common variables. In such systems, each equation represents a constraint that the solution variables must satisfy, and the goal is to find a set of values that meet all these requirements at once.

Matrix Representation

A system of linear equations can be succinctly represented in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the constant vector. This representation is a cornerstone in linear algebra, enabling the utilization of powerful matrix techniques to solve the system.

Matrix Inverse

An inverse of a square matrix A, denoted A?¹, is a matrix that, when multiplied by A, yields the identity matrix (A?¹A = I). The existence of an inverse indicates that the matrix is non-singular (i.e., it has a non-zero determinant), and it is crucial for solving systems of linear equations uniquely.

Solving Systems Using the Inverse Matrix

When a system of linear equations is expressed in the matrix form Ax = b, and A is invertible, multiplying both sides of the equation by A?¹ leads to the solution x = A?¹b. This method simplifies the process of finding solutions, particularly for complex numerical systems.