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When we talk about finding the roots of an equation, we're referring to the process of determining the values of the variable that make the equation true—in other words, where the equation equals zero. This is a fundamental aspect of algebra and is particularly important when dealing with functions and graphs. Finding roots is synonymous with solving an equation.

For a function such as \(f(x)\), the roots are the solutions to the equation \(f(x) = 0\). They represent the 'x-intercepts' or the points where the graph of the function crosses the x-axis on a coordinate plane. In the given example, setting \(f(x) = \frac{2x - 3}{7}\) equal to zero and solving for \(x\) reveals the root of the equation. Efficiently finding roots is a vital skill in mathematics as it paves the way for understanding more complex functions and equations.

Clearing the denominator is a technique used to simplify equations involving fractions, making it easier to solve for the unknown variable. This is done by multiplying both sides of the equation by the denominator to eliminate the fraction altogether.

In practice, if you have an equation like \(0 = \frac{2x - 3}{7}\), multiplying each side by 7, which is the denominator, will 'clear' it from the equation. It's crucial not to alter the balance of the equation, which is why both sides must be multiplied by the same number. After clearing the denominator, the equation becomes simpler, often reducing to a linear equation which is easier to solve. Clearing the denominator is essential in rational equations where managing fractions directly can be cumbersome.

Linear equations are the simplest type of equations that you'll encounter in algebra. They have the general form \(ax + b = 0\), where 'a' and 'b' are constants, and 'x' is the variable. Linear equations graph as straight lines, hence the term 'linear.'

These equations are straightforward to solve and generally involve basic operations: addition, subtraction, multiplication, and division. The ultimate goal is to isolate the variable on one side of the equation to find its value, as seen in the given exercise. Starting with \(0 = 2x - 3\), you add 3 to both sides to get \(3 = 2x\), and then divide by 2 to find that \(x = \frac{3}{2}\). The solution process showcases how even when starting with rational functions, the goal often boils down to solving a linear equation.

An algebraic fraction is simply a fraction where the numerator, the denominator, or both contain algebraic expressions. The key to working with algebraic fractions is the same as with numerical fractions—manipulate them in such a way as to find the values that make the equation true.

In the context of our problem, \(f(x) = \frac{2x - 3}{7}\) is an algebraic fraction. Similar to numerical fractions, we aim to simplify them by finding common denominators, canceling factors, or as previously mentioned, clearing the denominator altogether. This process is often necessary to solve equations or to simplify expressions in algebra. For example, we cleared the denominator to transform our original function into a more straightforward linear equation, which we could then solve to find our desired roots.