From \( -y = x \) and \( x = y \), substituting: \( -x = x \Rightarrow x = 0 \), then \( y = 0 \). Only solution is \(\mathbf{v} = egin{pmatrix} 0 \ 0 \end{pmatrix}\), but \( \|\mathbf{v}\|^2 = 0

["Solving the Linear System: The Unique Solution ( \mathbf{v} = \begin{pmatrix} 0 \ 0 \end{pmatrix} ) via Substitution", "Understanding the solution set of a system of linear equations is fundamental in linear algebra and has broad applications in engineering, physics, data science, and optimization. This article explores the derivation of the unique solution to the system defined by ( -y = x ) and ( x = y ), demonstrating how substitution yields ( x = 0 ) and ( y = 0 ), with the conclusion that the only solution is the zero vector — and its norm is zero.", "---", "### The System of Equations", "We begin with the two equations:\n[\n\begin{cases}\n-y = x \\nx = y\n\end{cases}\n]", "At first glance, these may appear redundant, but carefully substituting one into the other reveals the system’s structure and unique solution.", "---", "### Substitution Step: Replacing ( x ) with ( y )", "From the second equation,\n[\nx = y\n]", "Substitute this expression for ( x ) into the first equation:\n[\n-y = x \Rightarrow -y = y\n]", "Solve for ( y ):\n[\n- y = y \Rightarrow -y - y = 0 \Rightarrow -2y = 0 \Rightarrow y = 0\n]", "---", "### Back-Substitution to Find ( x ):\nSince ( x = y ), substituting ( y = 0 ) yields:\n[\nx = 0\n]", "Thus, the only pair satisfying both equations is:\n[\nx = 0, \quad y = 0\n]", "---", "### Identifying the Unique Solution", "Writing the solution as a vector:\n[\n\mathbf{v} = \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}\n]", "The squared Euclidean norm of ( \mathbf{v} ) is:\n[\n|\mathbf{v}|^2 = 0^2 + 0^2 = 0\n]", "This result confirms that the system has exactly one solution — the zero vector — and its norm is zero, reflecting geometric significance: the intersection point of the two lines occurs only at the origin.", "---", "### Why This Matters", "In linear systems, a unique solution like this emerges when the coefficient matrix of the system has full rank — meaning the equations are independent and consistent. The system ( -y = x ), ( x = y ) represents two intersecting lines at ( (0,0) ), a foundational concept in solving underdetermined or determined systems.", "Moreover, verifying that ( |\mathbf{v}|^2 = 0 ) reinforces the concept that norm zero vectors correspond to singular, balanced solutions — prevalent in least-squares problems and eigenanalysis.", "---", "### Conclusion", "From the simple substitution ( -y = x ) and ( x = y ), we logically deduce that the only solution is:\n[\n\mathbf{v} = \begin{pmatrix} 0 \ 0 \end{pmatrix}\n]\nwith norm:\n[\n|\mathbf{v}|^2 = 0\n]", "This elegant derivation underscores the power of substitution in linear algebra and highlights the meaningful geometric and algebraic insight embedded in seemingly simple equations.", "---", "Keywords:\n( -y = x ), ( x = y ), linear system, substitution, unique solution, zero vector, vector norm, linear algebra, solution derivation, ( \mathbf{v} = \begin{pmatrix} 0 \ 0 \end{pmatrix} ), ( |\mathbf{v}|^2 = 0 )"]