equation simplification. $(5y-1)/3 + 4 =(-8y+4)/6$ Simplification of this equation gives two answers when approched by two different methods.
Method 1 Using L.C.M( least common multiple)
$(5y-1)/3 + 4 =(-8y+4)/6$
$(5y-1+12)/3 = (-8y+4)/6$
$5y-11 =  (-8y+4)/2$
$(5y-11)2= (-8y+4)$
$10y-22 = -8y+4$
$18y=26$
$y = 26/18=13/9$
Method 2 multiplying every term by 3
$3(5y-1)/3 + 4*3 = 3(-8y+4)/6$
$5y-1 + 12 = (-8y+4)/2$
$2(5y-1 + 12) = -8y+4$
$10y-2+24  =  -8y+4$
$18y + 22 = 4$
$18y = -18$
$y = -1$
The correct method is method 2 and the correct answer is y = -1
Why is method 1 is incorrect? Could anyone explain why the answer is wrong when using the L.C.M( method 1)?
 A: Well, in the first case, there is a sign error:
$$\frac{5y-1+12}{3} = \frac{-8y+4}{6}$$
$$5y-11 = \frac{-8y+4}{2}$$
It should be 
$$5y+11 = \frac{-8y+4}{2}$$
A: Your first method multiplies  $(5y-1+12)/3$ by $3$ to give $5y-11$ when it should give  $5y+11$
A: We have $5y-1+12=5y+11$ and not $=5y-11.$
A: When you do two algebraic manipulations simultaneously, like here:
$$(5y-1+12)/3 = (-8y+4)/6$$
$$5y-11 = (-8y+4)/2$$
i.e. multiply through by $3$ AND simplify the left bracket, you have confusion. It should be $5y+11$. And you could always multiply through by $6$ instead of $3$ ($6=\operatorname{lcm}(3,6)$).
I would:
$$(5y-1)/3 + 4 =(-8y+4)/6$$
Multiply both sides by $6$:
$$2(5y-1)+24=-8y+4$$
Expand brackets:
$$10y-2+24=-8y+4$$
Simplify $-2+24$:
$$10y+22=-8y+4$$
Add $8y$ to both sides:
$$10y+22+8y=-8y+4+8y$$
Simplify:
$$18y+22=4$$
Add $-22$ to both sides
$$18y+22-22=4-22$$
Simplify:
$$18y=-18$$
Divide both sides by $18$:
$$y=-1$$
A: From the second step transitioning to the third step of your work, you incorrectly computed $5y-1+12$ to be $5y-11$ when it should be equal to $5y+11$. An obvious careless sign error.
A: They are both correct methods. (Actually they are so similar they really can't really be considered to be different methods.)
It's just that you did the second without making an arithmetical error, and you did the first with an arithmetical error.
$5y - 1 + 12 = 5y +11 \ne 5y-11$.
Fix that and every thing will work out fine.
