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Understanding the concept of the least common denominator (LCD) is crucial when solving linear equations involving fractions. The LCD is the smallest number that all denominators can divide into without leaving a remainder. Identifying the LCD helps to eliminate fractions and simplifies the equation.

For example, when dealing with fractions such as \(\frac{3}{5}\) and \(\frac{7}{10}\), we notice that the denominators are 5 and 10. The smallest number that both 5 and 10 can divide into without a remainder is 10. This means that 10 is our least common denominator. The strategy is to multiply each term by the LCD to transform the equation into one without fractions, making it easier to solve.

This process simplifies calculations and helps to clearly see the steps required to isolate variables effectively.

Isolating variables is a fundamental step in solving linear equations. The goal is to solve for an unknown variable, typically denoted as \(x\), by getting it on one side of the equation on its own. Once the fractions in an equation have been cleared using the LCD method, we usually end up with a more manageable equation.

Using our example, we transformed the original equation to \(10x + 6 = 7\) after eliminating fractions. The next step is to isolate \(x\) by performing inverse operations. We first subtract 6 from both sides to get \(10x = 1\) and then divide both sides by 10 to find that \(x = \frac{1}{10}\). Remember, the purpose of each operation is to reverse the effect of the numbers or operations applied to \(x\) for us to find its value.

Verifying solutions is the final, but essential step in solving equations. Once you believe you have solved the equation, you must check that the value obtained for \(x\) satisfies the original equation. Insert the value of \(x\) back into the initial equation and see if the equation holds true.

In this instance, substituting \(x = \frac{1}{10}\) into the original equation \(x + \frac{3}{5} = \frac{7}{10}\) gives us \(\frac{1}{10} + \frac{3}{5} = \frac{7}{10}\). To add the fractions on the left side, we find a common denominator, which in this case is again 10, leading to \(\frac{1}{10} + \frac{6}{10} = \frac{7}{10}\). Upon simplifying, we see that the left side equals the right side, confirming that our solution is indeed correct. Always perform this check to ensure accuracy in your solutions.