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When tackling a system of linear equations, the substitution method is a powerful tool. It involves substituting one equation into another. This reduces the system to a single equation with one variable, which is simpler to solve. Start by solving one of the equations for one variable. In our example, the first equation is already solved for y: \( y = -6x \). Next, substitute this expression into the other equation: \( 9x + 2(-6x) = 1 \).

By substituting, we convert a two-variable system into a single-variable equation. This makes it easier to solve for the unknown variable. Once you find the value of x, substitute it back into the first equation to find the value of y. Finally, always verify your solution by substituting the values back into the original equations to ensure they satisfy both. This method is systematic and helps clear up complexities in solving linear systems.

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. The standard form of a linear equation in two variables is \( ax + by = c \), where a, b, and c are constants. In our problem, the linear equations provided are: \( y = -6x \) and \( 9x + 2y = 1 \).

Linear equations can be graphed as straight lines on the Cartesian plane. The point where the lines intersect represents the solution to the system. This graphical representation is handy for visualizing solutions, though algebraic methods like substitution are typically more precise for exact solutions. Understanding linear equations is crucial for solving systems of equations, as it forms the basis of these methods.

A system of equations is a set of two or more equations with the same variables. The aim is to find values of variables that satisfy each equation simultaneously. In our example, the system consists of two linear equations: \( y = -6x \) and \( 9x + 2y = 1 \). There are different methods to solve these systems, such as graphing, substitution, and elimination. Each method aims to determine where the equations intersect, revealing the values that work for all given equations.

Systems of equations can have a single unique solution, infinite solutions (if the equations are dependent), or no solution (if the equations are inconsistent). In this problem, utilizing the substitution method reveals that the system has a unique solution: \( x = -\frac{1}{3} \) and \( y = 2 \). Always checking the solution by substituting back into the original equations is essential to confirm the solution's validity. Mastering solving systems of equations is fundamental in algebra and applicable in various practical and theoretical scenarios.