6/7S - 3/4 = 4/5 - 1/7S + 1/6

6/7S - 3/4 = 4/5 - 1/7S + 1/6

(6/7)s + (1/7)s = 4/5 + 1/6 + 3/4
s = 48/60 + 10/60 + 45/60
s = 103/60

Thanks so much Reiny!

To solve the equation 6/7S - 3/4 = 4/5 - 1/7S + 1/6, we need to follow certain steps to simplify and solve for the variable S.

Step 1: Combine like terms
First, let's combine the terms on the left side of the equation and the terms on the right side of the equation.

We have:
6/7S - 3/4 = 4/5 - 1/7S + 1/6

To combine the fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 7, 4, 5, and 6 is 420. So we need to convert the fractions to have a denominator of 420.

Multiplying 6/7S by 60/60, 3/4 by 105/105, 4/5 by 84/84, and 1/6 by 70/70, we get:

(360/420)S - (315/420) = (336/420) - (60/420)S + (70/420)

Now the equation becomes:

(360/420)S - (315/420) = (336/420) - (60/420)S + (70/420)

Step 2: Simplify the equation
Next, let's simplify the equation by performing the required arithmetic operations. Note that the denominators are equal, so we can eliminate them.

360S/420 - 315/420 = 336/420 - 60S/420 + 70/420

Dividing both numerator and denominator by 15, we get:

24S/28 - 21/28 = 24/30 - 4S/28 + 7/28

Simplifying further, we have:

24S/28 - 21/28 = 24/30 - 4S/28 + 7/28

24S - 21 = 24 - (4/28)S + 7

Step 3: Isolate the variable S
Next, let's isolate the variable S by moving all the terms with S to one side of the equation and all the constant terms to the other side.

24S + (4/28)S = 24 - 21 - 7

Simplifying the equation, we have:

(24 + 4/28)S = 24 - 21 - 7

(24 + 1/7)S = 24 - 21 - 7

Simplifying further, we get:

(24 + 1/7)S = 24 - 21 - 7

Now, let's simplify the expression (24 + 1/7):

(24 + 1/7) = (168/7 + 1/7) = 169/7

So the equation becomes:

(169/7)S = 24 - 21 - 7

Step 4: Solve for S
Finally, let's solve for S by dividing both sides of the equation by (169/7).

(169/7)S / (169/7) = (24 - 21 - 7) / (169/7)

Simplifying, we get:

S = (24 - 21 - 7) / (169/7)

Simplifying further, we have:

S = (24 - 28) / (169/7)

S = -4 / (169/7)

To divide by a fraction, we can multiply by its reciprocal:

S = -4 * (7/169)

Simplifying the expression, we get:

S = -28/169

Therefore, the value of S that satisfies the equation 6/7S - 3/4 = 4/5 - 1/7S + 1/6 is S = -28/169.