Question

How do you solve `\frac { v - 1} { 6} = 31- \frac { 6v - 1} { 2}`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the equation by `color(red)(6)` to eliminate the fractions while keeping the equation balanced:

`color(red)(6) xx (v - 1)/6 = color(red)(6)(31 - (6v - 1)/2)`

`cancel(color(red)(6)) xx (v - 1)/color(red)(cancel(color(black)(6))) = (color(red)(6) xx 31) - (color(red)(6) xx (6v - 1)/2)`

`v - 1 = 186 - (cancel(color(red)(6))3 xx (6v - 1)/color(red)(cancel(color(black)(2))))`

`v - 1 = 186 - 3(6v - 1)`

`v - 1 = 186 - (3 * 6v) - (3 xx - 1)`

`v - 1 = 186 - 18v - (-3)`

`v - 1 = 186 - 18v + 3`

`v - 1 = 186 + 3 - 18v`

`v - 1 = 189 - 18v`

Next, add `color(red)(1)` and `color(blue)(18v)` to each side of the equation to isolate the `v` term while keeping the equation balanced:

`v - 1 + color(red)(1) + color(blue)(18v) = 189 - 18v + color(red)(1) + color(blue)(18v)`

`1v + color(blue)(18v) - 1 + color(red)(1) = 189 + color(red)(1) - 18v + color(blue)(18v)`

`(1 + color(blue)(18))v - 0 = 190 - 0`

`19v = 190`

Now, divide each side of the equation by `color(red)(19)` to solve for `v` while keeping the equation balanced:

`(19v)/color(red)(19) = 190/color(red)(19)`

`(color(red)(cancel(color(black)(19)))v)/cancel(color(red)(19)) = 10`

`v = 10`