89. Solve for x in the equation:

6/7x - 6 = -2

I posted this before, and this is the reply I got from someone:

multiply both sides by 7
6x-42=-14
add 42 to each side
6x=28
x=14/3

But I don't get how this is done because to eliminate 6/7 shouldn't I multiply by 7/6, and not just 7?

Here's what I would do:

first, add 6 to both sides
6/7x=4
multiply both sides by 7/6
x=14/3

But I come out to the same answer..which process is correct?

Both are correct. Both do the same thing. Multiplying by 7 then dividing by six is the same thing as you did, multiplying by 7/6.

Both processes are correct, and they essentially follow the same steps with slight variations in notation. I'll explain both methods step-by-step to clarify why they yield the same answer.

Method 1 (as explained by the other person):
1. Multiply both sides of the equation by 7 to eliminate the fraction:
(7)(6/7x - 6) = (7)(-2)
6x - 42 = -14

2. Add 42 to both sides to isolate the variable:
6x = -14 + 42
6x = 28

3. Divide both sides by 6 to solve for x:
x = 28/6
x = 14/3

Method 2 (as you mentioned):
1. Add 6 to both sides of the equation:
6/7x + 6 = -2 + 6
6/7x + 6 = 4

2. Multiply both sides by 7/6 to eliminate the fraction:
(7/6)(6/7x + 6) = (7/6)(4)
x + 7/6(6) = 28/6

3. Simplify both sides:
x + 7/6 * 1 = 14/3

The 7/6 * 1 equals 7/6, so we have:
x + 7/6 = 14/3

4. Subtract 7/6 from both sides to isolate the variable:
x = 14/3 - 7/6

To subtract fractions, we need a common denominator, which in this case is 6:
x = (28/6 - 7/6) / 6
x = 21/6 / 6
x = 21/6 * 1/6
x = 21/36

5. Simplify the fraction:
x = 7/12

As you can see, both methods lead to the same final result of x = 14/3 or x = 7/12. The reason why both methods are correct is because multiplying both sides by 7 or multiplying by 7/6 is essentially performing the same operation – scaling both sides of the equation by a factor of 7 or 7/6 respectively.