91Ó°ÊÓ

In mathematics, a system of linear equations consists of two or more linear equations involving the same set of variables. Each equation can be represented graphically as a line. The goal is often to find the point where these lines intersect, which represents the solution to the system.

For example, consider a system with two variables, as shown in the original exercise:

• \( x_{1} + 2x_{2} = 4 \)
• \( x_{1} - x_{2} = 6 \)

Each equation corresponds to a line when graphed. If these lines intersect at one point, the system has a unique solution. If they coincide, there are infinitely many solutions. If they are parallel, there is no solution. Understanding how to determine the intersection is crucial in resolving these systems.

Determinants play a vital role in linear algebra when analyzing systems of linear equations. A determinant is a special number calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible or the volume transformation it describes.

For a system of linear equations, like our example, you can form a coefficient matrix and compute its determinant to see if there's a unique solution.

The coefficient matrix \( A \) from our exercise is:\[A = \begin{bmatrix} 1 & 2 \ 1 & -1 \end{bmatrix}\]Calculating the determinant of this matrix using the formula for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is:\[ \text{det}(A) = ad - bc = (1)(-1) - (2)(1) = -3 \]The determinant is non-zero, indicating the system has a unique solution where the lines intersect.

A coefficient matrix is a matrix created from the coefficients of the variables in a system of linear equations. It serves as a useful way to organize information and make calculations easier in linear algebra.

For our system:

• \( x_{1} + 2x_{2} = 4 \)
• \( x_{1} - x_{2} = 6 \)

The coefficient matrix becomes:\[A = \begin{bmatrix} 1 & 2 \ 1 & -1 \end{bmatrix}\]Each row corresponds to one equation, and each column corresponds to one variable. By calculating the determinant of the coefficient matrix, we determine whether the system has a unique solution. This matrix is integral in assessing the consistency and solvability of the system.

When discussing solutions to systems of linear equations, a unique solution means that there is exactly one point at which all equations in the system intersect. This occurs only when the determinant of the coefficient matrix is non-zero, as seen in the solution to our exercise example.

For the system:

• \( x_{1} + 2x_{2} = 4 \)
• \( x_{1} - x_{2} = 6 \)

The determinant of the coefficient matrix is \(-3\). Because this value is not zero, the system has a unique solution. In simple terms, the lines described by the equations are intersecting at exactly one point.

This unique intersection point is the solution to the system. Knowing how to identify whether a system has a unique solution is essential for solving linear equations and applying linear algebra concepts effectively.