91Ó°ÊÓ

When solving linear equations, a fundamental step is to isolate the variable. This is essentially getting the variable we're solving for by itself on one side of the equation, with no numbers or other variables attached to it. In the given exercise, \(1.205x - 0.003 = 0.5\), we aim to isolate \(x\).

This is done by performing the same mathematical operations to both sides of the equation, ensuring it remains balanced. To isolate \(x\), we add \(0.003\) to both sides, eliminating the \(0.003\) on the left and giving us a new, simplified equation: \(1.205x = 0.503\). The goal is to make the coefficient of \(x\), which is the number directly multiplying the variable, equal to 1. This makes the next steps to solving the equation simpler.

Algebraic operations are the bread and butter of solving equations. They include addition, subtraction, multiplication, division, and the distributive property, among others. Once our variable \(x\) is isolated with a coefficient, as we have in \(1.205x = 0.503\), the next step is to use algebraic operations to solve for \(x\).

In this case, we divide both sides of the equation by \(1.205\) to find the value of \(x\). This process, dividing by the coefficient, ensures that we are left with \(x\) on one side of the equation, achieving the goal of isolating the variable. It's essential to remember that whatever operation you perform on one side, you must also perform on the other to maintain the equation's balance.

In practical scenarios, especially when dealing with measurements, we often round decimals to make the numbers more manageable. In the provided exercise, it was specifically asked to round the answer to two decimal places, which means we want the number to have two digits after the decimal point.

After dividing \(0.503\) by \(1.205\) to isolate \(x\), we arrive at a decimal. To adhere to the instructions, we must round this decimal to two places. If the third decimal place is 5 or greater, we round up the second decimal place by one. If it is less than 5, we leave the second decimal place as it is. This rounding provides a more concise answer and follows the requirements of the given exercise. Rounding is a crucial skill in mathematics, as it aids in estimating and simplifies many problems.