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Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols typically represent numbers, and the rules dictate how these numbers can be combined and transformed. In the equation provided, we have numbers and the variable \( x \), which we need to solve for. The main objective in algebraic equations is to find the value of the unknown variable.

To solve an algebraic equation, we often use operations like addition, subtraction, multiplication, and division to isolate the variable. This process involves rearranging the equation to simplify it or to express it differently. In the original problem, the goal is to find the value of \( x \) that makes the equation true. By isolating \( x \) on one side of the equation, we can determine its value.

Rational expressions are fractions where the numerator and the denominator are both polynomial expressions. In our equation \( 9 = \frac{78}{x} \), the term \( \frac{78}{x} \) is a basic form of a rational expression. The key point to remember is that a rational expression is undefined if its denominator is zero.

This means it's crucial to ensure that \( x \) is not zero when manipulating these kinds of equations. For this particular exercise, we are solving for \( x \) to find when the equation balances out.

Handling rational expressions often requires you to eliminate the fraction. This can be done by multiplying every term in the equation by the least common denominator, which in this case is simply \( x \). By doing so, you remove the fraction and simplify the equation. This makes it easier to perform other operations and find the solution.

Multiplication is one of the core operations used in algebra to simplify and solve equations. In our exercise, we use multiplication to manipulate the equation \( 9 = \frac{78}{x} \). First, we multiply both sides by \( x \) to eliminate the fraction, resulting in \( 9x = 78 \).

This step clears the rational expression, making the equation easier to work with. Once the equation is in a simpler form, you can use division—another key operation—to solve for \( x \).

It’s important to remember that multiplication and division are inverse operations, so they are often used together in algebra to isolate a variable. In this case, dividing both sides by 9 gives us \( x = \frac{78}{9} \). This strategy of using multiplication to manage rational expressions is a powerful tool in algebra, helping students to tackle a wide array of problems.