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When working with a system of linear equations, one efficient way to find a solution is using the substitution method. This technique involves two primary steps: first, solve one of the equations for one of the variables, and then substitute this expression into the other equation.

The goal is to replace variables with equivalent expressions until you have an equation with just one variable. In the given exercise, the first step involved isolating the variable x in one of the equations. After finding x = 4y + 14, this expression for x is substituted into the other equation, effectively reducing the system to a single equation with one unknown, y. By solving for y, you're halfway to the solution; the final step is to plug the value back into the isolated equation to solve for x.

Utilizing this method can streamline the process of solving systems, especially with linear equations where variables can be easily isolated.

A system of linear equations consists of two or more linear equations containing two or more variables. The point where these equations intersect represents the solution set for the system. In our case, we have two equations: 2x + 3y = 6 and x - 4y = 14. These can graphically be represented as straight lines on a coordinate plane.

The solution to the system is the point that lies on both lines, meaning the x and y values at that point satisfy both equations. Systems can have one solution (intersect at a point), no solution (lines are parallel), or infinitely many solutions (lines are coincident). The steps shown in the solution illustrate the process of algebraically finding this point of intersection, leading to a single solution for this particular system.

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In the context of solving a system of equations, algebraic expressions are manipulated to isolate variables and solve for their values.

In the exercise, x was expressed in terms of y, creating an algebraic expression x = 4y + 14. This expression was then substituted into another equation, altering it to a form that could be solved for y. Understanding how to work with algebraic expressions is fundamental in algebra, enabling students to rearrange and combine equations to reveal the values of unknowns. Being adept in manipulating these expressions is crucial for solving equations efficiently, as showcased in the method used for this exercise.