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I need help solving -10(1/2x-1/5y)+30(1/6x+4/5y)

I need to simplify this expression
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6 Answers

This is Algebra I so I know you mean 1/2x as the fraction 1/2 times the variable x.
There are two sets of parenthesis where you need to distribute the outside factor. That means multiply each term inside the parenthesis by the outside factor.
1st work the left set of parenthesis
-10(1/2x - 1/5y)     In this section notice that the denominators 2 and 5 work well with the number 10 because they are factors of 10.
Multiply Each Term by -10    
 -10(1/2x) = -5x     and -10(1/5y) = -2y     Why? Because half of -10 equals -5 and a fifth of -10 equals -2.
Rewrite the expression  
The left parenthesis should now be simplified to -5x - (-2y)  and further to -5x + 2y because subtracting a negative is the same as adding a positive (integer subtraction rule).
2nd work the right set of parenthesis.
30(1/6x + 4/5y)  notice the denominators 6 and 5 work well with 30 because they are factors of 30.
Multiply each term by 30
30(1/6x) = 5x     and 30(4/5y) = 24y    
                                      Why 24? Because 30(1/5) = 6 but they are asking for 4/5 which is 4*6
Rewrite the right side of the expression.
5x + 24y
Now write them back together.     -5x + 2y + 5x + 24y    
Combine the like terms 
-5x + 5x  = 0 
and all that remains is 
2y + 24y = 26y    
26y is the simplified expression
The expression 1/2x is ambiguous because under the two versions of order of operations it could be (1/2)x under the modern interpretation or 1/(2x) under the classic interpretation because the classic interpretation holds that the product 2x takes precedence over the explicit division sign /.
Solution 1:  (classic):
multiply through using the distributive law to get
-5/x + 2/y + 5/x  +24/y
multiply the numerator and denominator of each expression by 2xy to get 
-10y/2xy + 4x/2xy + 10y/2xy + 48x/2xy    =  52x/2xy  
= 26/y   provided that x not = 0 and y not = 0.
Solution 2:  (modern)
multiply through using the distributive law to get
-5x + 2y + 5x + 24y   =   26y
So we get two different answers depending on the version of order of operations.   If we consider implied multiplication to have precedence we get 26/y;  if we consider implied and explicit multiplication to have the same precedence then we get 26y.   
Assuming you just need to simplify the expression, your best bet is to distribute. I'll do this term by term, then put it all together at the end. I'll assume that "1/2x" means "one half x", not "one over two x"
By distributing, I mean
   -10(1/2x-1/5y) = -10(1/2x) + (-10)(-1/5y)
= -5x - (-2y) = -5x + 2y.
This last equality is true because, for example, 10(1/2x) = 10(1/2)x = 10*(1/2)*x = 5x. This is only the first term. For the second, we distribute again and get
   30(1/6x +4/5y) = 30(1/6x) + 30(4/5y)
= 30(1/6)x + 30(4/5)y
= 5x + 4(6)y
= 5x + 24y.
Now we just have to add the two together, so 
   -10(1/2x-1/5y) + 30(1/6x+4/5y)
= (-5x + 2y) + (5x+24y)             (the simplifications we did before, added together)
= (-5 + 5)x + (2 + 24)y              (moving around parentheses)
= 0x +26y                                 (performing additions inside parentheses)
= 26y
So, our expression can be simplified to 26y.
I see comments revolving around the expression 1/2y, and related terms.
If it really was (1/2)*y, I would expect it to be expressed as y/2. That's how I would have set it up, anyway.
Kind regards,
Oops. Math error on first pass:
First, remove parentheses using the distributive principle:
a(b+c) = ab + ac
-10(1/2x - 1/5y) + 30(1/6x + 4/5y)
-10/2x + 10/5y + 30/6x + 120/5y
Next, multiply terms so that you have a common denominator:
For the first term, -10/2x, multiply the top and bottom by 3:
-30/6x + 10/5y + 30/6x + 120/5y
Combine like terms (x's and y's):
-30/6x + 30/6x + 10/5y + 120/5y
Perform addition:
And if this is what was meant:
-10(1/(2x)-1/(5y))+30(1/(6x)+4/(5y)) =
-5/x + 2/y +5/x + 24/y =
caveats: neither x nor y is zero.