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I need a convincing math problem or real-world example, to show that simplifying algebraic expressions such as $2x+7y-3+6y-9x-2$ is really important and profitable, and it's reasonable to learn how to do it.
Do you know any such problem or example?
P.S. The problem or example should be comprehensible for a student which has just got to know the algebraic expressions of degree one; no familiarity with graphing equations or using algebra for solving challenging problems.
IS there a number $x$ such that $2x-7−3x+6−9x−2+10x=0$ ? Is there a number $x$ such that $2x-7−3x+6−9x−2+11x=0$?
You can encapsulate these question in word problems easily. For example: "Alicia has received and made gifts in dollar and bitcoins: she got two bitcoins, then gave 7 dollars, then offered 3 bitcoins, then received 6 dollars, gave 9 bitcoins, gave 2 dollars, and last received 10 bitcoins. Brahimi says the bitcoin rate is unsufficient for her to break even. Is he right? At what bitcoin rate would she break even?"
The point of the particular two examples is that although they look very similar, their answers are very different. Once simplified, one sees immediately why.
The question seems to overlook the value of brevity in our writing, mathematical or otherwise. Shorter is better -- it makes more efficient use of our writing and time, makes things easier to read and understand, and also reduces possible errors (due to fewer parts to possibly mis-read or transcribe).
That might bear saying in a class once and should be a fairly obvious point. If one were in an English class and had overly verbose, rambling prose that needed editing down (while keeping the meaning the same), hopefully one would likewise see the value in that.
I suspect that finding a "real-world example" is unlikely, because this is something done in the pure algebraic expression alone. It does not reflect a change in any outside model or application.
Your restrictions are quite limiting. But never the less, for a linear expression, you can definitely do something.
Approach A
I have had my student write stories for an expression. So something in like $$ 3x + 4y - 5 $$ You could say that $x$ represents the number of oranges and $y$ the number of apples. The students could write something like:
Joanne sold $x$ oranges for \$3.00 each, $y$ apples for \$4.00 each, and \$5.00 were stolen from the till
and for a different expression
$$ 6x + 7y -3y +18 -2x -23 $$
the students could come up with a story like
Jim sold $x$ apples for \$6.00 each, $y$ oranges for \$7.00 each, bought another $y$ oranges for \$3.00 each, found \$18.00 lying on the ground, bought $x$ apples for \$2.00 each, and spent \$23.00 for a new kitchen table
So maybe they're hoakie. But now we can ask, "Who has more money? Jim or Joanne? Explain your answer."
There are all sorts of variations on this. What I'd stress here isn't an application of simplifying algebraic expressions, I'd stress that expressions can be tied to stories. The more students play with and create these sorts of stories, the less intimidating word problems are for them. But it does take a lot of work. And, it takes a lot of convincing to get them to realize what is and isn't a good story. At first, many of their stories will be nonsensical (I'm assuming you're teaching high school or junior high school students).
OK. So there's an application---contrived as it is---that you could use in your class. But, if you allow graphing and the students have spent some time working with quadratic equations, then you have a lot more room to show the importance of simplifying.
Approach B
Here the context is to talk about scientific exploration/inquiry. If you develop a theory, you might be inclined to express the relationship between two quantities in one form, but after collecting data, it might be more sensical to present a relation between the two quantities differently.
You can present a situation where two different people have come up with two different expressions:
Kate came up with the following equation for one set of data $$ y = (x+2)(x-4) $$ Ken had a different set of data and wrote out the following equation $$ y = (x-1)^2 - 9 $$ Now provided you've talked about graphing, zeroes of polynomials, and the vertex of a parabola, then you can talk about a context where simplifying can answer questions. You can ask, "Could the data Ken and Kate collected come from similar experiments?" Just looking at the data wouldn't suffice since the data sets are different. But, if the students know how to simplify the expressions, then they can answer the question.
Summary
Granted you might feel you want a clean example when first presenting the notion of gathering like-terms and simplying expressions. But sometimes, you just don't have enough to work with yet to make the point. It's important to teach your students that there isn't always an immediate answer to why we do something. But then promise them that you'll revisit the idea in a few weeks when they've developed some more skills and a few more ideas. If you're good to your promises, in my experience, the students will often play along.
In all of this, long range planning is important. You need to know what skills and knowledge base you're starting with in your students; how you plan to build on that; and, where you intend to get. Sometimes, to create interesting and convincing "real world" applications, you have to be creative in how you present the material in the first place.
How about counting proteins, fats, and carbs? For example, 100 g of bread contain
total fat - 3 g, including saturated fat - 1 g
total carbs - 51 g, including fiber - 2 g, sugars - 4 g
proteins - 8 g
The nutritional value of 1 g of fat is 9 kcal, 1 g of protein - 4 kcal, 1 g of carbs but not dietary fiber - 4 kcal, dietary fiber - 0.
Suppose that $x$ is the weight of one piece of bread in grams. You eat two pieces of bread for breakfast, three pieces of bread for lunch, and one as an afternoon snack. What's the amount of fats, sugars, non-fiber carbs and the total nutritional value of all the bread you eat in the whole day?
Later, when you introduce systems of linear equations, you can pose questions like planning daily menu out of, say, bread, eggs and salmon so that the total calorie intake is 2000 with 50% coming from carbs, 30% coming from protein and 20% coming from fats.
