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

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3 Answers

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!

Comment

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

Ignore the 4w since you won't do anything with it now. You will pick it up later. Then distribute the -9 with the contents of the parentheses. Since a negative times a positive equals a negative, -9 times 6= -54 and since a negative times a negative equals a positive, -9 times - 5w equals positive 45w.  Next copy the 4w and the = sign and the 44 so it becomes

4w - 54 + 45w = 44 

Add 54 to both sides of the equal sign. 54 is cancelled from the left side of the equation, but is added to the 44. It becomes

4w -54 +54 +45w  = 44+ 54.  Combine 4w and 45w so now the equation becomes 4w + 45w= 98

49w = 98  To get the w by itself, divide each side of the equation by 49

49w/49 =98/49 

49 cancels 49 so the w stands alone. 98 divided by 49 equals 2

so the answer is 

w = 2

 

 

Danielle,

The key to solving this problem is to understand the distributive property.

This property allows us to multiply numbers into groups of numbers. For example, 

if we had the expression 5*(x+6), this property allows us to multiply the 5 through the group of numbers consisting of x and 6.

So, lets use this property to make our problem easier to solve.

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

we can use the distributive property to simplify the part that reads -9(6-5w).

To do this we take (-9) and multiply it by each piece inside the parenthesis : (-9)(6) +(-9)(-5w)

This leaves us with -54 + 45w. Now we simply put this back into our equation.

4w -54 + 45w = 44

with some rearranging we get :

49w - 54 = 44

next, we bring the 54 to the other side by adding it to both sides. Remember, whatever you do to one side of and equation you must do to the other side. 

we get 49w = 98.

Finally, we divide both sides by 49 to get us to the final answer of w = 2.

Hope this is helpful.