I have no idea how to figure this out... please help.

Danielle,

This question tests your knowledge of the **distributive **property of algebra.

Symbolically, the **distributive **property can be explained as follows:

*x**(*y* + *z*) = (*x***y*) + (*x***z*) = xy + xz, or alternatively

*x**(*y - z*) = (x**y*)* -* (*x***z*)

...where *x*,* y*, and *z* are numbers (real, rational, etc.). In words, when you multiply a number by the sum or difference of a group of terms,

x * (*y + z*)

then you can convert this into the sum or difference of *x* "times" each of those individual terms. On the right hand side of the first equation,

x*y + x*z

You see how the "+" symbol stays in place, and both *y* and *z* are multiplied by *x* separately. Then you just have to add the two terms to get your answer! (Of course, you must first perform the multiplication of *x*y *and *x*z*
to know what you are adding.)

So let's use the **distributive** property to solve your question, which I will restate:

4w - 9(6 - 5w) = 44

Ignoring the "4w" term for a moment, let's use the distributive property to expand the "9(6 - 5w)". Using what we learned above, you want to multiply 9 by 6, place your "-" sign after it, then multiply 9 by 5w. This will give us

9(6 - 5w) = (9*6) - (9 * 5w)

Note that the expression on the right side is now a difference of two terms, which after multiplication, can be simplified to

(9*6) - (9 * 5w) = (54) - (45w) = 54 - 45w

And *w*, as you may know, is a variable in this case...it should be treated like a regular number (so any operation can be performed on it, like multiplication), but you don't know it's value yet.
Thus 9*5w is the same as 9*5*w, which is 45*w = 45w.

Anyway, after that expansion using the distributive property, you have

4w - (54 - 45w) = 44

Which is another way of writing

4w + (-1)*(54 - 45w) = 44

because when you subtract a number, it is the same process as adding that number "times" -1.

So we use the distributive property AGAIN to get

4w + (-1*54) - (-1*45w) = 4w + (-54) - (-45w) = 4w - 54 + 45w

Now I will state your initial question one final time, given that I found an easier way to express "4w - 9(6 - 5w)" :

4w - 54 + 45w = 44

At this point you combine "like terms", so on the left hand side of the equation you would add 4w and 45w to get 49w.

(4w + 45w) - 54 = 44 ---> 49w - 54 = 44

Next you add 54 to both sides of the equation

49w - 54 + 54 = 44 + 54

49w = 98

As a final step, you divide both sides by 49, to get *w* alone on one side (which gives you the value of *w*!)

49 * w / 49 = 98 / 49

(49 / 49) * w = (98 / 49)

1 * w = 2

w = 2

And there you have it... the solution to this problem is w = 2.

Hope this helps! And if anything in this explanation needs clarification, don't be afraid to ask!

## Comments

Great job, Joseph! Students really need to see it written out the long way. I was about to answer, but you did it as I would. Thorough!