How to expand a term using the associative property?

siruku

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Hi there... first post! :) Thanks to anyone who can take a look... prepping up for math and am covering some gap areas.

I am working on multi-step equation:
plot-formula.mpl

So I use this site to solve (because I'm not sure how to solve it. :confused:)

On step 6:
plot-formula.mpl


It says to "expand term 1/2 using the associative property on {a + 2}" What does that even mean? :(
The resulting picture is:
plot-formula.mpl


I'm guessing its saying that the fractions encircled by green are actually equivalent to "a + 2 / 2", e.g.: 1 * a / 2 + 1* 2/2 == a + 2 / 2

Can someone clarify what it means by this? (And how one might draw the same conclusion?) I looked up the term and didn't find anything which seemed relevant to how its being used here. Thank you for any assistance!

Steps 1-6:
associative_property.jpg
 

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I am working on multi-step equation:

plot-formula.mpl


On step 6:

plot-formula.mpl
I can't read the complete list of steps, but it seems like too many steps. They have done four things, going from the first equation above to the second.

They used the commutative property of addition to rewrite (2+a) as (a+2). They used the fact that multiplication by 1/2 is the same as division by 2. They evaluated 3^2 and got 9. They subtracted (3a+4)/9 from each side.


It says to "expand term 1/2 using the associative property on {a + 2}" What does that even mean?
The statement does not make sense, to me.

Maybe they were trying to describe rewriting (expanding) the rational expression (a+2)/2 as a sum of the rational expressions a/2 + 2/2. Yet, they could have acheived that at the beginning, by using the distributive property.

Other methods are more direct. We can eliminate the ratios in the original equation, using the least common multiple (LCM) of the denominators 2 and 9.

Here's how it works. We multiply each side of the original equation by 18 (LCM), and both denominators cancel.

\(\displaystyle \displaystyle \frac{18}{1} \cdot \frac{1}{2} \cdot (2 + a) = \frac{3a + 4}{9} \cdot \frac{18}{1}\)

Simplify.

9(2 + a) = 2(3a + 4)

Now we expand each side, using the distributive property.

18 + 9a = 6a + 8

We collect like-terms. Subtract 18 from each side, and subtract 6a from each side.

3a = -10

Multiply each side by 1/3, to solve for a.

a = -10/3 :cool:
 
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I just noticed the link in your thread. My advice: Stop using that calculator! :shock:

It took 20 steps, to explain how it first converted the given equation to:

a/6 + 5/9 = 0

Then (as shown below), it switched to decimal representations for 1/6 and 5/9 (reported to 15 places, ha). The site doesn't actually show the final steps, to isolate symbol a, and it reports an approximation for the answer instead of the exact answer.

For the given equation, I think that site's list of steps is garbage.


CLICK IMAGE TO ENLARGE

garbage.JPG
 
I just noticed the link in your thread. My advice: Stop using that calculator! :shock:

It took 20 steps, to explain how it first converted the given equation to:

a/6 + 5/9 = 0

Then (as shown below), it switched to decimal representations for 1/6 and 5/9 (reported to 15 places, ha). The site doesn't actually show the final steps, to isolate symbol a, and it reports an approximation for the answer instead of the exact answer.

For the given equation, I think that site's list of steps is garbage.


CLICK IMAGE TO ENLARGE

View attachment 9839

Thank you for the responses, this definitely helps! I mostly use the site to double-check my answers, if an answer isn't provided, or to see (early) if I am on the right track.

I have a question on the simplification section of your answer... when you multiply both sides by the LCM, how does (3a + 4 / 9) * (18 /1) become 2(3a + 4) ? Or, at least what is this method called, so I can read into it more?
 
I mostly use [that calculating] site to double-check my answers, if an answer isn't provided, or to see (early) if I am on the right track.
Using technology to check an answer is usually okay, but relying on lists of steps to see whether you're going in the right direction is not. First, it will dull your mind (over time) because you're letting a machine do your thinking for you. Second, there are often multiple ways of solving the same equation, so you might not receive good direction. (Your site is a prime example of this.)

One may also check their own solution candidate(s). Substitute your answer back into the original equation, and evaluate each side (to be sure they're equal).


… when you multiply both sides by the LCM, how does (3a + 4 / 9) * (18 /1) become 2(3a + 4) ? …
9 goes into 18 two times. So, the fraction 18/9 reduces to 2.

Can you see what's happening, in the following, simpler example?

\(\displaystyle \dfrac{x}{9} \cdot \dfrac{18}{1} = \dfrac{x \cdot 18}{9 \cdot 1} = \dfrac{x \cdot 2}{1} = 2x\)
 
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I hesitate to provide links to online calculators, but not in this case, because that other site you're using upsets me.

Try this site, instead.

Use an asterisk for multiplication. Also, as you type your expressions, pay attention to the position of the insertion point (i.e., the blinking cursor). When you type a ratio, the insertion point will move into the denominator position. After typing the denominator, use the right-arrow key to move out of the denominator position. Alternatively, you can click on the templates provided at the top.
 
... when you multiply both sides by the LCM,
how does (3a + 4 / 9) * (18 /1) become 2(3a + 4) ?
That should be [(3a + 4) / 9] * (18 / 1),
which is same as (3a + 4) / 9 * 18 ; OK?
 
I'm glad Denis noted the issue with grouping symbols in (3a + 4 / 9)

Those outer ( ) are not needed. We need grouping symbols around the numerator, to show that the entire number (3a + 4) is being divided by 9, not just the 4.

When we type algebraic ratios in calculators and other math software, we especially need to be mindful to use grouping symbols correctly.

Like this: 1/2*(2 + a) = (3*a + 4)/9

If you tell a machine 3a + 4 / 9 it will follow the Order of Operations and interpret the expression as 3a + (4/9) -- that is, 3a is no longer being divided by 9.

To see examples of how to type math with a keyboard (and to find math symbols), the forum guidelines provide links to web sites for help. :cool:
 
I am working on multi-step equation:
plot-formula.mpl
Like Mark, I shiver at how complicated "they" made it seem...
Can be this simple:

Step1: rearrange slightly
(a + 2) / 2 = (3a + 4) / 9

Step2: crisscross multiplication
9(a + 2) = 2(3a + 4)
So:
9a + 18 = 6a + 8
3a = -10
a = -10/3
 
Using technology to check an answer is usually okay, but relying on lists of steps to see whether you're going in the right direction is not. First, it will dull your mind (over time) because you're letting a machine do your thinking for you. Second, there are often multiple ways of solving the same equation, so you might not receive good direction. (Your site is a prime example of this.)
Understood! I will usually try to solve it by hand first, and if I feel stuck... then I use the calculator, which is much more immediate than asking someone else. Forcing myself to do it by hand has been a challenge (where historically I just wouldn't do it all all), as I abhor the rote approach despite enjoying, say the idea of solving it, haha.
One may also check their own solution candidate(s). Substitute your answer back into the original equation, and evaluate each side (to be sure they're equal).


9 goes into 18 two times. So, the fraction 18/9 reduces to 2.

Can you see what's happening, in the following, simpler example?

\(\displaystyle \dfrac{x}{9} \cdot \dfrac{18}{1} = \dfrac{x \cdot 18}{9 \cdot 1} = \dfrac{x \cdot 2}{1} = 2x\)

This makes sense. I believe I've been getting hung up on the idea of a denominator for a number that "doesn't exist" e.g. the variable.
 
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