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Understanding the system of linear equations is fundamental in algebra. It consists of two or more linear equations with the same set of variables. Our goal is to find the common solution that satisfies all equations simultaneously.

For instance, in our exercise, we're working with a system of two equations: \(y = \frac{1}{4}x - 5\) and \(2x - \frac{3}{2}y = 1\). These equations are like puzzle pieces that fit together in just one specific way, which in this case is the point where the two lines intersect. By finding the values of \(x\) and \(y\) that work for both equations, we are essentially locating the point on a graph where the two lines cross each other. This point is unique for each pair of lines, provided that the lines are not parallel.

We often represent these equations graphically, with each equation forming a line. The intersection point, when it exists, is where the lines meet.

The term algebraic methods encompasses a variety of techniques used to solve equations, including those in a system of linear equations. Some well-known methods include substitution, elimination, and graphing. Each method has its own advantages depending on the particular system.

In the exercise, we used an algebraic approach to determine the point of intersection. After we arranged our equations in a convenient way, we were able to manipulate them algebraically to find a common solution. These techniques are not just a series of random steps but require a strong understanding of algebraic principles such as distribution, combining like terms, and inverses. Mastering these algebraic methods is critical as these are tools you'll use to crack many types of mathematical problems.

The substitution method is particularly useful when one equation in the system is already solved for a variable. It involves substituting the expression for that variable from one equation into the other equation. This method takes the value of one variable in terms of the other variable and replaces it, hence the name 'substitution'.

In the exercise, we first isolated \(y\) in the equation \(y = \frac{1}{4}x - 5\) and then replaced \(y\) in the second equation with the expression found. This step transforms the system of equations into a single equation in one variable, in our case, \(x\). The substitution method requires careful algebraic manipulation and is highly effective when dealing with linear systems.

The final stage of our exercise is solving equations, which is where the 'magic' happens. After substituting one variable, we're left with an equation in one variable, which we need to solve. Solving an equation means finding the value of the variable that makes the equation true.

In the exercise, we manipulated the equation step by step to isolate \(x\), using arithmetic operations such as distributing, combining like terms, and applying inverses (addition and subtraction). Once we found the value of \(x\), we plugged it back into one of the original equations to find \(y\). Solving equations is a foundational skill in algebra that you'll use over and over, whether you're working on simple single-variable problems or tackling complex multivariable systems.