Question

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

Answer 1

`x=-2`

Explanation:

`(x+2)/9-(2x+9)/5=2x+3`

Multiplying first term by `5/5` and the second by `9/9` on the right side of the equation to get a common denominator:
`((x+2)*5)/(9*5)-((2x+9)*9)/(5*9)=2x+3`
`(5x+10)/45-(18x+81)/45=2x+3`
`((5x+10)-(18x+81))/45=2x+3`

Simple simplification:
`(5x+10-18x-81)/45=2x+3`

Then we multiply both sides of the equation by 45 to get rid of denominator on the left side of the equation:
`(5x+10-18x-81)/cancel(45)*cancel(45)=(2x+3)*45`
`5x+10-18x-81=90x+135`

Grouping like-terms next to each other and then evaluating them:
`5x-18x+10-81=90x+135`
`-13x-71=90x+135`

Adding 13x to both sides of the equation to get the terms with x in it to one side:
`cancel(13x)-71+cancel(13x)=90x+135+13x`
`-71=103x+135`

Then we subtract 135 from both sides to isolate the term that has the unknown x in it:
`-71-135=103xcancel(+135)-cancel(135)`
`-206=103x`

Then we divide both sides of equation by 103 to isolate x:
`(cancel(-206)-2)/cancel(103)=(cancel(103)x)/cancel103`
`-2=x`

`rarr color(red)(x=-2)`