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The substitution method is a popular technique for solving systems of equations, especially when dealing with linear equations. This approach involves substituting one variable in terms of another, allowing us to solve the system in a step-by-step manner.

To begin using the substitution method, you first need to isolate one of the variables in one of the equations. In the original exercise, the second equation, \(2x + y = 9\), was chosen to isolate \(y\). By rearranging the equation, we find \(y = 9 - 2x\).

Once you have expressed one variable in terms of the other, substitute this expression into the other equation. This transforms the system into a single equation with one variable, making it easier to solve. In our example, substituting \(y = 9 - 2x\) into the first equation \(3x - 5y = 7\), we can then solve for \(x\). After achieving this, substitute back to find the value of the other variable, \(y\). This two-step process simplifies solving systems by reducing the number of unknowns.

The substitution method is particularly useful when one of the equations is easily solved for a single variable. It provides clarity by eliminating one variable at a time, simplifying your work dramatically.

Linear equations are a fundamental part of algebra. They represent relationships where the degree of each term is one, hence forming a straight line when graphed.

In the system given, both equations are linear, illustrated as follows:

• \(3x - 5y = 7\)
• \(2x + y = 9\)

Each equation consists of two variables, \(x\) and \(y\), and all terms are first-degree. This means we are dealing with a two-dimensional line in the Cartesian plane.

The main objective when solving linear equations in a system is to find the point where these lines intersect. This point of intersection represents the values of \(x\) and \(y\) that satisfy both equations simultaneously. With the substitution method, we reduced the complexity by handling one linear equation at a time, solving first for \(x\) and then\(y\). Therefore, understanding linear equations are crucial as they form the foundation of many real-life applications, including economic models and engineering principles. By simplifying linear equations, algebra provides a pathway to predict outcomes and make sense of data.

Solving algebraic equations involves finding the variable values that make the equations true. The objective is to simplify each equation and isolate the variables through various operations such as addition, subtraction, multiplication, or division.

In the exercise provided, after substituting \(y = 9 - 2x\) into the first equation, we focused on solving for \(x\). This required expanding and regrouping terms to form a solvable expression:

1. Expand to get \(3x - 45 + 10x = 7\).
2. Combine similar terms, resulting in \(13x = 52\).
3. Finally, divide both sides by 13, yielding \(x = 4\).

With \(x\) determined, we substitute back to solve for \(y\) in \(y = 9 - 2x\), which gives \(y = 1\). Each step is a vital part of solving algebraic equations, reinforcing the importance of careful manipulation and solving step-by-step.