91Ó°ÊÓ

In algebra, variable isolation is a foundational skill. It means rearranging an equation to have the variable on one side of the equation and constants on the other. For example, consider the equation given in the exercise, \(3.95 - x = 2.32x + 2.00\). To isolate \(x\), you need all the \(x\)-terms on one side. To accomplish this, start by adding \(x\) to both sides. This step adjusts the equation as \(3.95 = 2.32x + 2.00 + x\). Notice how the \(x\) initiates grouping responsibility on one side. Simplifying the terms \(2.32x + x\) gives \(3.32x\), leading to the simplified appearance: \(3.95 = 3.32x + 2.00\). By strategically handling the variable \(x\) and associated terms, variable isolation is achieved successfully.

Equation solving involves the process of finding the value of the variable that makes the equation true. Once you have isolated the variable as further demonstrated with \(3.95 = 3.32x + 2.00\), the next target is simplifying further to pinpoint \(x\). Begin by subtracting 2.00 from both sides: \(3.95 - 2.00 = 3.32x\). This transformation leads to \(1.95 = 3.32x\). Finally, solve for \(x\) by dividing both sides by 3.32, yielding \(x = \frac{1.95}{3.32}\). Solving provides \(x \approx 0.587\). Through these strategic actions via addition, subtraction, and division, equation solving clears pathways to the variable's exact value. Patience and accuracy are key in these steps.

After solving the equation and obtaining the variable value as a decimal, often you need to round it to a specified degree of precision. In this example, \(x \approx 0.587\) needs rounding to two decimal places. For rounding, observe the digit immediately after the second decimal place. If it is 5 or greater, increase the second decimal digit by one. Otherwise, leave it unchanged. Here, since the third place is 7 (greater than 5), \(0.587\) rounds to \(0.59\). Rounding numbers is essential for simplification in reporting and can often enhance clarity and communicate precision in your results.