A dialogue on a simple equation

 

T: My friends, let me see if you can solve the equation 2(x + 5) - (x - 4) = 5x - 19.

A: What do you mean “solve”?

T: To find all values of x that makes the left side equal to the right side.

A: So x stands for a number?

T: Yes.

A: Why don’t you write it like my last teacher did 2(• + 5) - ( • - 4) = 5• - 19?

T: OK.

 B: x is 6.

T: Great! We have a first solution. Can you find more solutions?

A: How did you find the solution?

B: I just guessed. 6 was the best guess I could think of.

A: So you didn’t test it?

B: Let me do it now. 2(6 + 5) - (6 - 4) = 2 * 11 - 2 = 20 and 5 * 6 - 19 = 11. Almost the same!

T: But not the same, so 6 is close, but not a solution.

B: Try 10 then. That should do it.

A: 2(10 + 5) - (10 - 4) = 2 * 15 - 6 = 24 and 5 * 10 - 19 = 31. Sorry B, wrong again! You should really stop guessing. Math is about thinking, not about guessing.

T: I like to disagree A. Guessing in math is excellent if it is followed by thinking. If you think a bit you can use B’s two wrong guesses to come up with a solution.

A: How?

T: That’s a very good question.

C: Miss, I think I have found a solution.

T: Excellent, please show us how you found it.

C: A few weeks ago we studied geometry, so when I saw the equation I didn’t think about numbers, but rectangles. There are three rectangles. One with width 2 and length x + 5, and two with width 1 and length x - 4 and 5x - 19 respectively.

A: How do you know the length of each rectangle. For instance, are you sure that 5x - 19 is longer than x + 5?

C: It has to be longer since x - 4 is obviously shorter than x + 5 and the two bottom rectangles have to equal the top one.

T: That’s a nice drawing C!

C: Since x - 4 is 9 less than x + 5 the rectangle with length 5x - 19 has to be split into two rectangles. One with length 9 and one with length 5x - 28.

A: But how does this help you to find x? Remember the problem was to find x not to draw beautiful rectangles!

C: Since the top side has to equal the bottom side x + 5 = 5x - 28. A few guesses told me that x = 8.25 makes the left side equal to the right side.

 T: I am impressed C!

D: I have another idea T.

T: Let’s hear it!

D: In my last school we did everything backwards.

A: What on earth do you mean?

D: When we solved equations we started with the answer and changed multiplications to divisions, additions to subtractions and so on.

T: Can you give a simple example, please?

D: If (x + 5) / 3 = 7 we multiplied 7 by 3 to get 21 and finally subtracted 5 to get 16 for x.

T: OK.

D: So, I wondered if I could solve the equation you gave us backwards as well. I made this drawing.

A: What a beautiful drawing!?

T: Can you please explain D?

D: I started by putting x in the square. 2(x + 5) means “add 5 to x and then times the answer by 2.” That is shown in the middle of the drawing. Similarly x - 4 is shown at the top and 5x - 19 at the bottom.

A: Why do you have circles with - in them?

D: The result of (x - 4) has to be subtracted from 2(x + 5). The result is put in the top circle with – in it. The arrow indicated which number is subtracted from which.

T: OK. I think I understand the drawing, but how do you use it to find x?

D: The last subtraction equals 0.

A: Really, I thought you had an oxygen atom in there with a double bound!

D: Very funny A! As I said, the last subtraction equals 0 so the two circles that lead into it have the same value which I call c.

B: I see. And c + 19 is backwards of subtracting 19?

D: Exactly. And when you divide by 5 you are back to x, so x = (c + 19) / 5. The top half is similar. The only difference is that since two circles have difference c I give them the values d + c and d respectively.

T: Again, I am impressed, but how does this lead to x?

D: At the left I have three equations for x. The first one tells me that d = x - 4 and the second that 
c = 5x - 19. When I put these values into the middle expression I get that x = (x - 4 + 5x - 19) / 2 - 5 or, to make it simpler, x = (6x - 23) / 2 - 5. Which is the same as x = 3x - 16.5. If you try x = 8.25 on the right side you get 8.25 making the two sides equal!

B: I am perplexed! I didn’t know two different methods could give the same answer! What is wrong teach?

T: With you or with the methods?!

B: Don’t be smart, Miss, sarcasm may set its marks.

T: Sorry! How many ways are there to Rome, B?

B: I don’t know.

T: Always one more than those you know. Of course a problem can be solved in zillions of ways. The question is, which is the best method.

B: And which is the best?

T: That’s a very good question!

A: I have found another way!

T: Let’s hear it.

A: I used B’s silly guesses from earlier. While you’ve been talking I have asked myself which of the two guesses was the best one.

B: So they were not that silly after all?

A: In the beginning I couldn’t decide. The guess that x = 6 gave left side = 20 and right side = 11 while x = 10 gave left = 24 and right = 31. In the first one the difference between the sides is 9 while in the second it is 7.

B: So the second is obviously the best!?

A: Yes, but the question is where is the correct solution hiding? I tried x = 8 and came up with this table. 

 B: What a beautiful table!

A: I noticed that when x increases by 2 the difference decreases by 8. For x = 8 the difference is 1 so I just have to decrease it by 1 more. 1 is 1/8 of 8 and 1/8 of 2 is 1/4, so the answer has to be x = 8.25!

T: What an excellent line of reasoning! This must be my best class! We now have three ways of solving the equation!

B: But Miss, be honest, you knew all of this already didn’t you?

T: Not at all! I can’t even count.

A: Be serious please!

T: OK. Let me show you how I solved the equation.

A: Great! This must be good!

T: 2(x + 5) - (x - 4) = 5x - 19 was the starting point.

I didn’t like the (x + 5) bit so I replaced it with y. In fact, I replaced all x’s with y - 5.

That gave me 2y - (y - 9) = 5y - 44.

I didn’t like the (y - 9) bit, so I replaced it with x. Actually I replaced all y’s with z + 9.

It led me to 2z + 18 - z = 5z + 1.

A: How did you like that equation?

T: That one I liked quite well, so I solved it. I found that z = 4.25.

A: But you had to find x, not z?

T: I know. y was 9 more than z, or 13.25, and x was 5 less than y, so x = 8.25.

B: I am amazed that after so much nonsense your landed on your feet! Why on earth did you solve it like that?

T: I wanted to test an old Chinese idea called fan-fa. It is sometimes good for solving more difficult equations, so I wanted to take it for a spin.

B: And what do you think? Do you like your method?

T: Not a lot! What you have found is much better.

C: Miss, I have found another way of solving the problem.

T: Go ahead please, we are all ears!

C: Last week I downloaded a shareware program from http://www8.pair.com/ksoft/ called graphmatica. I asked it to plot the left and the right side.

 

B: Look at that, the lines seem to intersect somewhere between 5 and 10!

C: Than I had the brilliant idea of plotting the difference between the two sides instead. That is, I plotted y = 2(x + 5) - (x - 4) - (5x - 19).

It is obvious from the graph that x = 8.25 is the solution.

T: Interesting C! Good job!

A: I had an idea! The backward reasoning inspired me to think if we, starting from x = 8.25, could create the original equation.

B: What on earth do you mean? Why construct the equation we already have?

A: Let me try to explain with an example. What is the solution to x - 27.25 = -19?

B: I have no idea!

A: 8.25, of course! All I did was to subtract 27.25 on both sides. If left side was equal to the right side before I subtract, of course, they will be the same afterwards.

B: What’s you point?

A: x - 27.25 = -19 looks a bit more like the original equation, we have the –19 in place.

B: And now you could add 5x to each side to get 6x - 27.25 = 5x - 19 and the right side is in place!

A: You got it!

C: There is one snag my friend! How do you know x = 8.25 to begin with? You don’t!

A: OK, so let’s work forwards then, start with the original equation and transform it to equations that are simpler, but that have the same solutions as the original.

B: You mean by adding and subtracting the same on each side?

A: Yes, and multiplying and diving if that is needed.

B: Sounds very complicated to me, but maybe it is a good idea.

T: Please try your method on the board A.

A: OK. Here goes.

Original equation: 2(x + 5) - (x - 4) = 5x - 19

First I multiply out the brackets: 2x + 10 - x + 4 = 5x - 19

Then I simplify each side individually: x + 14 = 5x - 19

I subtract 5x from each side: -4x + 14 = -19

I subtract 14 from each side: -4x = -33

I divide each side by –4: x = 8.25

C: That was neat! I wonder if it always works.

B: I have improved A’s method where he used my two initial guesses. The method I have found works for any equation!

T: Wow! Are you sure?

B: If I want f(x) = 2(x + 5) - (x - 4) - (5x - 19) to equal 0, all I have to do is to find two wrong guesses! One that gives a number too big and one that gives a number too small. At the start of this lesson I found that f(6) = 9 and that f(10) = -7.

C: But this is nothing new B! Come to the point, will you?!

B: If 6 is too much and 10 is too little, the answer has to be between 6 and 10!

C: Duh! Listen to Einstein everyone!

B: Being unprejudiced, my next guess is the number midway between 6 and 10. f(8) turns out to be 1, so now I know that the solution is between 8 and 10.

C: We have known that for some time already!

B: I know, but my method works for any equation! I’ll continue to take the midpoint till I get the solution.

C: And how many times is that? Won’t your method take forever?

B: I’ll use a computer of course. In a spreadsheet it may look like this. Only four steps were needed with my initial ‘silly’ guesses. 

 C: How did you make the spreadsheet? What’s under the hood?

B: Here are the formulas. 

 T: I am surprised C. What a show!

D: Miss, this class always gives me a headache. We have found six or seven ways to solve the equation, isn’t one enough?

T: Of course one is enough.

D: Which one is the best?

A: I liked the backward reasoning.

B: I liked A’s forward reasoning.

C: I liked the graph.

D: Which one is the best?

T: That’s a very good question. What are some criteria for a good solution?

B: It should be correct, always give the right answer.

A: It should be easy to use and easy to understand.

C: What are you saying? I don’t need to understand it if it is easy to use!

B: It should work on a whole set of similar problems.

D: It should serve as a tool to solve problems we haven’t met yet.

A: I should be able to program the method on my calculator or computer.

D: I like to be able to do it by hand, without depending on a computer.

B: It should be neat or elegant.

C: Be quiet everyone, I would like Miss to tell us which one is the best. But first, always when we ask you a really important question you just say “That’s a very good question.” Why is that? Don’t you want to help us?

T: That’s a very good question.