To simplify this expression, we need to find a common denominator for each term.
The common denominator for the first term, 6c/7, and the second term, 5d/9, is 63. Therefore, we multiply the first term by 9/9 and the second term by 7/7:
(6c/7) * (9/9) = 54c/63
(5d/9) * (7/7) = 35d/63
Now, for the third term, 1c/2, and the fourth term, 1d/3, the common denominator is 6. Therefore, we multiply the third term by 3/3 and the fourth term by 2/2:
(1c/2) * (3/3) = 3c/6
(1d/3) * (2/2) = 2d/6
Combining these simplified terms, we have:
54c/63 - 35d/63 - 3c/6 + 2d/6
Next, let's find a common denominator for all the terms, which is 63 * 6 = 378. Therefore, we multiply each term by an appropriate factor to make the denominators 378:
(54c/63) * (6/6) = 324c/378
(35d/63) * (6/6) = 210d/378
(3c/6) * (63/63) = 189c/378
(2d/6) * (63/63) = 126d/378
Now, we can combine the terms:
324c/378 - 210d/378 - 189c/378 + 126d/378
Combining the like terms, we get:
(324c - 189c)/378 - (210d - 126d)/378
Simplifying the numerators:
135c/378 - 84d/378
Finally, we can simplify the expression further by dividing both terms by their greatest common divisor, which is 9:
(135c/9) / (378/9) - (84d/9) / (378/9)
15c/42 - 56d/42
Now, we can simplify the expression further by dividing both terms by their greatest common divisor, which is 7:
(15c/7) / (42/7) - (56d/7) / (42/7)
2c/6 - 8d/6
Simplifying the numerator further:
c/3 - 4d/3
Therefore, the simplified expression is c/3 - 4d/3.
Simplify 6c/7 - 5d/9 - 1c/2 + 1d/3
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