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Multiplication is one of the basic operations in mathematics, often used to find the total number when you have groups of the same size. In the context of linear equations, multiplication serves to balance or unbalance the equation. For instance, the equation \[ 23,465 \times x = 8,142,355 \], was initially multiplied by 23,465 to produce a larger number, \ 8,142,355\.

In such equations, the term multiplied is called the coefficient (23,465 in this case), and the resulting product is another number. When solving, we usually aim to 'undo' this multiplication to isolate the variable (x). Understanding multiplication helps demystify why certain steps are taken to solve such equations.

Division is the inverse operation of multiplication and is crucial in solving linear equations where a variable is being multiplied by a number. To isolate the variable, you perform division.

Take the given equation, \[ 23,465 \times x = 8,142,355 \], the multiplication must be 'undone' by dividing both sides by the coefficient, 23,465.

When you divide both sides, you get: \[ x = \frac{8,142,355}{23,465} \].

This step is key in simplifying the equation and finding the value of x. Understanding division helps you see why we divide both sides of the equation by the same number to maintain balance.

Isolating the variable is the goal when solving a linear equation. This means getting the variable (x in this case) alone on one side of the equation.

Following the steps for the given equation \[ 23,465 \times x = 8,142,355 \]:

- First, identify and focus on the variable you need to isolate (here, it's x).
- Next, determine how the variable is being manipulated (here, it's being multiplied by 23,465).
- Use the inverse operation (division in this case) to 'undo' the multiplication. This simplifies to: \[ x = \frac{8,142,355}{23,465} \].

By performing these steps, you isolate x and simplify the problem to a straightforward calculation, yielding x = 347. Understanding how to isolate the variable ensures you can tackle a wide range of linear equations effectively.