To solve this equation for ‘y’ you need to rearrange the equation so that ‘y’ becomes the subject of the equation – you will get an equation that states ‘y=(the rest of the equation)’. To do this you need to step by step rearange the terms that are currently on the ‘y’ side of the equation so that ‘y’ can be isolated.

Remember that when rearranging algebra equations the first rule is that ‘what you do on one side of the equation, you must also do on the other side of the equation’ this is because the equation must remain balanced.

The equation is:

`5x-2y-1=0`

Step 1: the first (and easiest) term of the equation that can be rearranged is the ‘-1’. We want to ‘get rid’ of the -1 from the ‘y’ side of the equation, but still keep the equation balanced. Adding +1 to the ‘y’ side of the equation will cancel the -1 out (because -1+1=0). Then because of the rule ‘what you do on one side of the equation, you must also do on the other side of the equation’ we also need to add +1 to the other side.

`5x-2y-1+1=0+1`

`5x-2y=1`

Step 2: the next term to rearrange is the ‘5x’. We want to ‘get rid’ of the 5x from the ‘y’ side of the equation, but still keep the equation balanced. Subtracting -5x from the ‘y’ side of the equation will cancel the 5x out (because 5x-5x=0 regardless of the actual value of x). Then because of the rule ‘what you do on one side of the equation, you must also do on the other side of the equation’ we also need to place a -5x on the other side.

`+5x-5x-2y=-5x+1`

`-2y=-5x+1`

Step 3: now we just need to isolate the ‘y’ by ‘getting rid’ of the -2 from the ‘y’ side of the equation. The –2 is not a term of the equation, it is a coefficient of y so is treated differently. The full term ‘-2y’ means that -2 is being multiplied with ‘y’ so to ‘get rid’ of the -2 we need to divided by -2. (Because -2/-2=1 and any number multiplied by 1 is still the same number 1y=y). Then because of the rule ‘what you do on one side of the equation, you must also do on the other side of the equation’ we also need to divide the other side of the equation by -2. (NOTE: all terms on the other side of the equation need to be divided by -2)

`(-2y)/(-2)=(-5x)/(-2)+(1)/(-2)`

`y=-(5x)/(-2)+(1)/(-2)`

Step 4: now to tidy up the equation. `(-5x)/(-2)` is a negative divided by a negative so the negatives cancel each other out. Then `(5x)/(2)` can be more correctly written as the fraction `(5)/(2)x` ; and `(1)/(-2)` can be more correctly written as -`–(1)/(2)` .

`y=(5)/(2)x-(1)/(2)`

So the quick version, solving for y, would look like this:

`5x-2y-1=0`

`5x-2y=1`

`-2y=-5x+1`

`y=(-5x)/(-2)+(1)/(-2)`

`y=(5)/(2)x-(1)/(2)`

5x - 2y -1 =0

( add 2y to both sides)

5x - 1 = 2y

( divide both sides by 2)

(5/2)x - 1/2 = y

or y = (5/2)x - 1/2