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The substitution method is a commonly used technique in solving systems of linear equations. It is especially convenient when one of the variables is easily isolated.
The core idea is to solve one of these equations for a variable, and then substitute this expression into the other equation. This method simplifies the problem, reducing the system of equations to one single linear equation with only one variable.
This approach is particularly useful in scenarios where the equations are relatively simple and can easily be transformed. It allows you to find the solution systematically by breaking down the problem into steps. Here, for instance, isolating 'y' and substituting it in the first equation provided an immediate path to solve for 'x'.
The aim is to reduce the complexity by focusing on one variable at a time before putting it all back together.

Linear equations form the building blocks for solving most algebraic problems. A linear equation is a type of equation that makes a straight line when you graph it. It typically has variables raised to the power of one and has no variables in the denominator. It can be expressed in the general form: \[ ax + by = c \]
These equations represent relationships where there is a constant rate of change. In our example, \(3x - 5y = 7\) and \(2x + y = 9\) are the equations we want to solve.
Understanding the properties of linear equations is crucial because these properties help us form strategies, such as substitution or elimination, to find the solutions. They illustrate the relationship between variables in a straightforward and predictable manner.
Working with linear equations often involves finding the points of intersection, which are the solutions that satisfy both equations simultaneously.

Solving systems of linear equations through algebraic steps involves a sequence of operations that follow a logical path to the solution. The steps ensure you handle complex arithmetic and variable manipulation correctly.
Key steps of an algebraic solution typically include:

• Isolating a variable: Start by isolating one variable in one of the equations. For instance, solving for \(y\) in \(2x + y = 9\) gives \(y = 9 - 2x\).
• Substitution: Substitute the expression obtained into the other equation, replacing the isolated variable.
• Simplification: This step involves simplifying the resulting equation, combining like terms, and making calculations more manageable.
• Solving: Proceed by solving the simplified equation for one of the variables. Once found, plug this value back into the expression for the isolated variable to find the second variable's value.

These steps allow you to methodically derive both variable values accurately. Each step leads you closer to the final solution, ensuring nothing is overlooked.