Question

How do you solve and check your solutions to `8/5x=4/15`?

Answer 1

See a solution process below:

Explanation:

Multiply each side of the equation by `color(red)(5)/color(blue)(8)` to solve for `x` while keeping the equation balanced:

`color(red)(5)/color(blue)(8) xx 8/5x = color(red)(5)/color(blue)(8) xx 4/15`

`cancel(color(red)(5))/cancel(color(blue)(8)) xx color(blue)(cancel(color(black)(8)))/color(red)(cancel(color(black)(5)))x = (cancel(color(red)(5))color(red)(1))/(cancel(color(blue)(8))color(blue)(2)) xx (color(blue)(cancel(color(black)(4)))1)/(color(red)(cancel(color(black)(15)))3)`

`x = color(red)(1)/color(blue)(2) xx 1/3`

`x = 1/6`

To check the solution substitute `1/6` for `x` in the original equation and ensure both sides of the equation are equal:

`8/5x = 4/15` becomes:

`8/5 xx 1/6 = 4/15`

`(8 xx 1)/(5 xx 6) = 4/15`

`8/30 = 4/15`

`(2 xx 4)/(2 xx 15) = 4/15`

`(color(red)(cancel(color(black)(2))) xx 4)/(color(red)(cancel(color(black)(2))) xx 15) = 4/15`

`4/15 = 4/15`

Answer 2

Solution: `\frac{1}{6}`

Check: see below.

Explanation:

Any equation of the form

`ax=b`

yields the solution

`x = b/a`

In your case, `a = 8/5` and `b = 4/15`

So, we must divide by `a`, which is a fraction. So, `b/a` is the same thing as `b \cdot a^{-1}`, where `a^{-1}` is the inverse fraction of `a`, obtained by switching numerator and denominator. So, the solution is

`x = b/a = \frac{4/15}{8/5} = \frac{cancel(4)}{cancel(15)3}\cdot\frac{cancel(5)}{cancel(8)2} = 1/6`

To check the value, simply plug it in, substituting `x`:

`\frac{8}{5}\color(green)(x) = \frac{4}{15} \implies \frac{cancel(8)4}{5}\color(green)(\frac{1}{cancel(6)3}) = \frac{4}{15}`

which is true, since the left hand side evaluates to `\frac{4}{15}`