Determine LCM(12,9,18)=square Using the LCM to clear the denominators of -(7x)/(12)+(5)/(9)=(1)/(18) we get the following equation: square =square . -(7x)/(12)+(5)/(9)=(1)/(18) has solution: x.=square

To determine the least common multiple (LCM) of 12, 9, and 18, we first find the prime factorizations of each number:<br /><br />- 12 = 2^2 * 3<br />- 9 = 3^2<br />- 18 = 2 * 3^2<br /><br />The LCM is found by taking the highest power of each prime that appears in the factorizations:<br /><br />- The highest power of 2 is 2^2.<br />- The highest power of 3 is 3^2.<br /><br />Thus, the LCM is 2^2 * 3^2 = 4 * 9 = 36.<br /><br />So, $LCM(12,9,18)=36$.<br /><br />Next, we use the LCM to clear the denominators of the equation $-\frac{7x}{12} + \frac{5}{9} = \frac{1}{18}$.<br /><br />Multiplying each term by the LCM of 36, we get:<br /><br />$36 \left(-\frac{7x}{12}\right) + 36 \left(\frac{5}{9}\right) = 36 \left(\frac{1}{18}\right)$<br /><br />Simplifying each term:<br /><br />$-3 \cdot 7x + 4 \cdot 5 = 2$<br /><br />$-21x + 20 = 2$<br /><br />Solving for $x$:<br /><br />$-21x = 2 - 20$<br /><br />$-21x = -18$<br /><br />$x = \frac{-18}{-21}$<br /><br />$x = \frac{6}{7}$<br /><br />Therefore, the solution to the equation $-\frac{7x}{12} + \frac{5}{9} = \frac{1}{18}$ is $x = \frac{6}{7}$.