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When solving systems of linear equations, dealing with fractions can be challenging. We can make calculations easier by eliminating the fractions. To do this, we use the Least Common Multiple (LCM) of the denominators. This process converts the fractions into whole numbers. Consider the equation: \(\frac{1}{2} x + \frac{1}{3} y = 3\). Here, the denominators are 2 and 3. To eliminate the fractions, we multiply both sides of the equation by the LCM of 2 and 3, which is 6. Doing this will transform the equation into a simpler form: 6 * \frac{1}{2} x + 6 * \frac{1}{3} y = 6 * 3\(3x + 2y = 18\).Similarly, for the second equation \(\frac{1}{4} x - \frac{2}{3} y = -1\), with denominators 4 and 3, we use the LCM of 12: 12 * \frac{1}{4} x - 12 * \frac{2}{3} y = 12 * (-1)\(3x - 8y = -12\). By eliminating fractions, the equations involve only whole numbers, simplifying further steps.

The LCM is crucial in solving equations with fractions. It's the smallest number that multiple denominators can divide into without leaving a remainder. Finding the LCM helps convert fractions into whole numbers. To find the LCM of two numbers:

• List the multiples of each number.
• Identify the smallest multiple common to both lists.

For example, to find the LCM of 4 and 3, start listing the multiples: Multiples of 4: 4, 8, 12, 16, ...Multiples of 3: 3, 6, 9, 12, ...The smallest common multiple is 12, so the LCM of 4 and 3 is 12. Using the LCM helps simplify equations and makes computations more manageable.

An inconsistent system of linear equations has no solutions. This occurs when the lines represented by the equations do not intersect. To determine if a system is inconsistent, follow these steps:

• Reduce the equations to a simpler form.
• Compare the simplified equations.

If the simplified equations contradict each other, the system is inconsistent. For instance, suppose after eliminating fractions and simplifying, we get the following: \(3x + 2y = 18\)\(3x + 2y = 22\).These equations suggest that the same linear combination of \(x\) and \(y\) equals two different numbers, which is impossible. Hence, the system is inconsistent. Recognizing and identifying an inconsistent system early on can save time and help you understand the nature of the problem you're working with.