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Solving an equation means finding the value of a variable that makes the equation true. For any equation, our goal is to isolate that variable on one side. This process involves simplifying and manipulating the equation carefully.

To effectively solve equations, follow these basic steps:

• First, simplify each side of the equation if possible by clearing any fractions or decimals.
• Use orders of operations to manage additions, subtractions, divisions, or multiplications.
• Isolate the variable, aiming to have it on its own on one side of the equation.

This approach ensures clarity and helps discern the path to finding the solution.

Decimals can make equations look complicated, but there's a straightforward way to clear them. By multiplying the entire equation by a power of 10, you can transform decimals into whole numbers, making calculations easier.

In our example, the equation involved decimals such as 0.06 and 0.1. To clear these decimals, identify the decimal with the highest number of places. Since 0.06 has two decimal places, multiply the entire equation by 100 (which is \(10^2\)):\[100(0.06x + 0.1(x - 300)) = 100(98)\]Executing the multiplication removes the decimals, converting it to a more manageable form: \(6x + 10(x - 300) = 9800\). This method is a vital skill for simplifying equations with decimals.

After clearing decimals and simplifying fractions, the next step is to combine like terms. Like terms refer to terms in an equation that contain the same variable to the same power.

Consider this simplified equation: \(6x + 10x - 3000 = 9800\). The terms \(6x\) and \(10x\) are like terms because they both involve the variable \(x\). By combining them, we simplify the equation:\[6x + 10x = 16x\]So, the equation adjusts to: \(16x - 3000 = 9800\). This step is crucial, as combining like terms simplifies the equation, making it easier to isolate and solve for \(x\).

Multiplication within equations involves distributing multiplication operations across terms, especially when handling parentheses.

Take the equation \(10(x - 300)\) from our example. The multiplication here requires distributing the 10 to both terms inside the parentheses:

• \(10 \times x = 10x\)
• \(10 \times (-300) = -3000\)

This distribution helps maintain equality while breaking down complex terms. Once distributed, the equation \(6x + 10(x - 300) = 9800\) transforms to \(6x + 10x - 3000 = 9800\).

Mastering multiplication and distribution is essential for expertly managing equations and guiding them towards a solution.