Question

How do you solve `\frac{2}{x + 3} + \frac{3}{2} = \frac{19}{10}`?

Answer 1

Using first principles

`x=2`

Explanation:

Given:`" "2/(x+3)+3/2=19/10`

Subtract `3/2` from both sides.

`2/(x+3)+0" "=" "19/10-3/2" "=" "19/10-15/10" "=" "2/5`

Cross multiply`color(green)( larr" Shortcut method. See foot note")`

`2xx5=2(x+3)`

Divide each side by 2

`5=x+3`

Subtract 3 from both sides

`x=2`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("Foot note")`

`color(purple)("This is why the much faster cross multiply method works:")`

As an example, suppose we had `2/x=3/y`

Multiply both sides by `color(blue)(x)`

`" "color(brown)(2/xcolor(blue)(xx x)" " =" "color(blue)(x xx)3/y`

`" "color(brown)(2xx (color(blue)(x))/x" " =" "(3color(blue)(x))/y`

But `x/x=1`

`" "2" "=" "(3x)/y`

Multiply both sides by `color(blue)(y)`

`" "color(brown)(2color(blue)(xx y)" "=" "(3x)/ycolor(blue)(xxy)`

`" "color(brown)(2color(blue)( y)" "=" "3x xx(color(blue)(y))/y`

But `y/y=1` giving:

`" "2y=3x`