Set this equal to \(\begin{pmatrix} 5 \\ 7 \\ -4 \end{pmatrix}\) and solve the system:
["Setting Up and Solving a System of Equations: An Example with a Vector Equation", "In linear algebra, one common task is to represent and solve systems of equations using vectors and matrices. These tools are essential in fields like computer graphics, engineering, and data science. In this article, we’ll explore how to set up a system of equations based on a vector equation and solve it step by step.", "---", "### Set the Equation:\nWe begin with a vector equation of the form:\n[\n\begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 5 \ 7 \ -4 \end{pmatrix}\n]\nThis equation asserts that the column vector (\begin{pmatrix} x \ y \ z \end{pmatrix}) is exactly equal to the given constant vector.", "---", "### Translating to a System of Equations\nTo solve for the unknowns (x), (y), and (z), we equate corresponding components:\n[\nx = 5\n]\n[\ny = 7\n]\n[\nz = -4\n]", "This is a simple system of three equations in three variables, where each equation corresponds to one component of the vector.", "---", "### Solving the System", "Because each equation involves only one variable, the solution is immediate:\n- From (x = 5), we get (x = 5).\n- From (y = 7), we get (y = 7).\n- From (z = -4), we get (z = -4).", "So the solution to the system is:\n[\nx = 5, \quad y = 7, \quad z = -4\n]\nIn vector form, the solution can be written as:\n[\n\begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 5 \ 7 \ -4 \end{pmatrix}\n]", "---", "### Why This Matters\nThis example illustrates a direct vector solution—where the vector is fully specified rather than derived from multiple equations. However, in more complex systems, matrices and vector spaces allow us to solve for unknowns through techniques like Gaussian elimination or matrix inversion. Setting such vector equations is foundational in modeling constraints, transformations, and data structure in mathematics and applied sciences.", "---", "### Conclusion\nSolving systems like (\begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 5 \ 7 \ -4 \end{pmatrix}) is straightforward when each variable corresponds to a single known value. Understanding this basic setup prepares you for more advanced applications involving linear systems, optimization, and beyond.", "---", "Keywords: vector equation, solve system of equations, linear algebra, matrix solver, setup vector equations, elementary linear systems, solve for variables, matrix and vector basics.", "---", "Practice Tip: Try replacing the constants with variables and solve for each variable individually—this strengthens your understanding of direct vector assignments and component-wise solving."]
