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A system of linear equations is essentially a collection of equations that require solving simultaneously because they share variables. The solution is a set of values that satisfy all the equations in the system. For example:

• Equation 1: last year's population of all age groups: \( x + y + z = 5525 \)
• Equation 2: this year’s population changes: \( 1.1x + 0.8y + 2z = 6040 \)
• Equation 3: relationship between age groups: \( y = x + 500 \)

These equations describe different relationships and constraints. To solve them and find the values of \(x\), \(y\), and \(z\), Cramer's Rule can be applied, given these equations are linear and consistent. This method involves using determinants, which form the backbone of Cramer's Rule.

The determinant is a special number that can be calculated from a square matrix. In the context of linear algebra, it helps us determine whether a system has a unique solution, infinitely many solutions, or no solutions at all.

• If the determinant is non-zero, the system has a unique solution.
• If it is zero, the system might have infinitely many solutions or none at all.

In this exercise, the main determinant \(D\) is derived from the coefficient matrix:\(D = \begin{vmatrix} 2 & 1.9 \0 & 2 \end{vmatrix} = 4.8\)Since \(D\) is non-zero, it guarantees the system has a unique solution, and Cramer's Rule can be employed to find specific values for \(x\), \(y\), and \(z\). The calculations for determinants like \(D_x\) and \(D_z\) follow a similar matrix manipulation, indicating the solution for each variable.

Matrix form is an efficient way to organize and handle systems of linear equations. Each equation corresponds to a row in the matrix, with coefficients and constants clearly aligned. For linear systems, this forms a structure that computers can process efficiently.To convert a system of equations to matrix form, you write it as:\[A\mathbf{x} = \mathbf{b}\]Where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the results vector.For this problem, the system of equations is represented in matrix form as:\[\begin{bmatrix} 2 & 1.9 & 0 \0 & 0 & 2 \end{bmatrix}\begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 5025 \6040 \end{bmatrix}\]This structured form makes it straightforward to apply linear algebra techniques, such as Cramer's Rule, to find solutions efficiently.

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It is essential for understanding and solving problems that involve linear relationships between variables. Key concepts of linear algebra include:

• Vectors: An ordered list of numbers, often representing points or directions in space.
• Matrices: Rectangular arrays of numbers that can represent systems of linear equations.
• Determinants: Derived from matrices; important for assessing the solvability of linear systems.
• Vector Spaces: Set of vectors that can be scaled and added together; these obey certain axioms.
• Linear Transformations: Operations that transform one vector into another, preserving the operations of addition and scalar multiplication.

In the context of solving the system of linear equations from the exercise, linear algebra provides the rules and tools, such as matrices and determinants, to find solutions using methods like Cramer's Rule.