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The substitution method is a technique used to solve simultaneous linear equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. This strategy effectively reduces a system of equations with multiple variables to a single equation with one variable, making it simpler to solve.

Let's take the provided exercise as an example. With the equation \(0.5x + 3.2y = 9.0\), we can solve for \(x\) to get an expression in terms of \(y\) which is \(x = 18 - 6.4y\). By substituting this into the second equation, we create an equation with a single variable \(y\), allowing us to find its value. Once we have the value of \(y\), we substitute it back into the isolated variable equation to find the value of \(x\). This step-by-step process simplifies the solution and ensures that you work methodically towards the correct answer.

Isolating a variable is crucial in solving equations and is the foundation of the substitution method. To isolate a variable means to manipulate the equation in such a way that you get the targeted variable on one side of the equation and all other terms on the other side.

In the given problem, we start by isolating \(x\) in the first equation. Isolation is achieved by moving all terms involving \(y\) to the right side and keeping \(x\) alone on the left side. This step not only prepares us for substitution but also helps to see the relationship between the variables at a glance. Being comfortable with moving terms across the equals sign through addition, subtraction, multiplication, or division is necessary for this process.

Simultaneous equations are a set of equations containing multiple variables that are all true at the same time. The goal when solving simultaneous equations is to find the values of the variables that satisfy all the equations in the system at once. Systems can be composed of two or more equations, and the strategies to solve them include substitution, elimination, and graphical methods.

In our problem, we have two simultaneous equations that characterize a linear relationship between \(x\) and \(y\). To solve these equations, we use the substitution method, which is ideal when one equation can be conveniently solved for one variable. This approach often provides a clear path to the algebraic solution for both variables.

Algebraic solutions involve finding the exact numerical answers to equations using algebraic manipulations. Unlike graphical solutions, which might only give an approximate answer based on where lines intersect on a graph, algebraic solutions provide precise values.

In our exercise, algebraic solution entails precisely determining the values of \(x\) and \(y\) that satisfy both equations simultaneously. After isolating \(x\), substituting it into the second equation, and solving for \(y\), we acquire an exact numerical value, which is then used to compute the exact value of \(x\). Verifying the calculated values by plugging them back into the original equations to ensure they are true solutions is a critical final step in confirming the algebraic solution.