Question

How do you solve `\frac{4v + 9}{5} = \frac{2v - 6}{3} + 5`?

Answer 1

See a solution process below:

Explanation:

First, eliminate the fractions by multiplying each side of the equation by the lowest common denominator of the two fractions which will also keep the equation balanced:

`color(red)(15)((4v + 9)/5) = color(red)(15)((2v - 6)/3 + 5)`

`cancel(color(red)(15))3((4v + 9)/color(red)(cancel(color(black)(5)))) = (color(red)(15) * (2v - 6)/3) + (color(red)(15) * 5)`

`3(4v + 9) = (cancel(color(red)(15))5 * (2v - 6)/color(red)(cancel(color(black)(3)))) + 75`

`(3 * 4v) + (3 * 9) = 5(2v - 6) + 75`

`12v + 27 = (5 * 2v) - (5 * 6) + 75`

`12v + 27 = 10v - 30 + 75`

`12v + 27 = 10v + 45`

Next, subtract `color(red)(27)` and `color(blue)(10v)` from both sides of the equation to isolate the `v` term while keeping the equation balanced:

`-color(blue)(10v) + 12v + 27 - color(red)(27) = -color(blue)(10v) + 10v + 45 - color(red)(27)`

`(-color(blue)(10) + 12)v + 0 = 0 + 18`

`2v = 18`

Now, divide each side of the equation by `color(red)(2)` to solve for `v` while keeping the equation balanced:

`(2v)/color(red)(2) = 18/color(red)(2)`

`(color(red)(cancel(color(black)(2)))v)/cancel(color(red)(2)) = 9`

`v = 9`