SOLUTION: How is (-3/4(-8a)+(-3/4)(-12) equivalent to both -3/4(-8a-12) and 6a+9?

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Question 1204943: How is (-3/4(-8a)+(-3/4)(-12) equivalent to both -3/4(-8a-12) and 6a+9?
Found 4 solutions by ikleyn, MathLover1, greenestamps, math_tutor2020:
Answer by ikleyn(53937) About Me  (Show Source):
You can put this solution on YOUR website!
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How is (-3/4(-8a)+(-3/4)(-12) equivalent to both -3/4(-8a-12) and 6a+9?
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In this post, the parentheses are unbalanced, so the formula is  FATALLY  UNREADABLE  and  FATALLY  INCORRECT.


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@MathLover1 solved  DIFFERENT  equation for you - different from what is written in your post,
even without warning/informing.

Which is a mathematical  CRIME.



Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
%28-3%2F4%29%28-8a%29%2B%28-3%2F4%29%28-12%29........-3%2F4 is common factor, so factor it out
=%28-3%2F4%29%28%28-8a%29%2B%28-12%29%29
=%28-3%2F4%29%28-8a-12%29=> proven to be equivalent to %28-3%2F4%29%28-8a-12%29+

now we can continue and multiply by -3%2F4
%28-3%2F4%29%28-8a-12%29+
=-%283%2F4%29%28-8a%29-%28-3%2F4%29%2812%29 ...simplify
=-%283%2F1%29%28-2a%29-%28-3%2F1%29%283%29+
=3%2A2a%2B3%2A3
=6a%2B9 => proven to be equivalent to+6a%2B9



Answer by greenestamps(13367) About Me  (Show Source):
You can put this solution on YOUR website!


The wording of your post implies that you don't understand why "-3/4(-8a-12)" and "6a+9" can both be equivalent to the given expression.

The reason is that those two expressions are equivalent:

%28-3%2F4%29%28-8a-12%29=%28-3%2F4%29%28-8a%29%2B%28-3%2F4%29%28-12%29=6a%2B9

Answer by math_tutor2020(3838) About Me  (Show Source):
You can put this solution on YOUR website!

You should be careful about the parenthesis placement.
If in doubt, use a calculator or CAS (computer algebra system) to validate the input.
A rule of thumb: There should be the same number of opening parenthesis "(" compared to the number of closing parenthesis ")". Otherwise things are unbalanced.

It appears you're asking how is (-3/4)(-8a)+(-3/4)(-12) equivalent to both (-3/4)(-8a-12) and 6a+9

The simple answer is distributive property
p(q+r) = p*q + p*r
Multiply the outer 'p' with each term inside.

For example,
2(3+4) = 2*3+2*4 = 6+8 = 14
and using PEMDAS we find that
2(3+4) = 2*(7) = 14
This is one example using numbers to verify the distributive property works.

Another example
3*(103)
= 3*(100+3)
= 3*100 + 3*3
= 300 + 9
= 309
In short, 3*103 = 309

One more example with numbers only
7*(215)
= 7*(200+10+5)
= 7*200 + 7*10 + 7*5
= 1400 + 70 + 35
= 1400 + 105
= 1505
In short, 7*215 = 1505

Now let's look at a few examples involving variables
4*(3x+5) = 4*3x + 4*5 = 12x + 20
and
7w*(9w+2) = 7w*9w + 7w*2 = 63w^2 + 14w
and
11(3+6p) = 11*3+11*6p = 33+66p = 66p+33
I encourage you to try other examples on your own.

Why can we extend the distributive property from numbers only to variables? Because variables are placeholders for numbers. It's a more abstract version.