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### Solving Problems With and Without Algebra

```Date: 05/14/2002 at 23:26:42
From: M Hanlon
Subject: Math equation

In a basketball game, Harold and Isaac scored a total of 19 points.
Isaac and Jacob scored a total of 14 points.  Isaac scored as many
ponts as Harold and Jacob together.  How many points did each player
score?

I have not taken algebra yet and do not know how to do this
equation.  I used various methods but could not come up with the
```

```
Date: 05/17/2002 at 12:43:17
From: Doctor Ian
Subject: Re: Math equation

Hi,

Just for fun, let me show you how you would solve the problem 'by
algebra'.  Then I'll show you how to solve it without algebra.

If we let H stand for Harold's points, and so on, we get

H + I = 19

I + J = 14

I = H + J

which is a set of three equations with three unknowns.  Note that
since I is the sum of H and J, we can just get rid of it by
substituting (H+J) wherever we see I:

H + (H+J) = 19

(H+J) + J = 14

(H+J) = H + J

The third equation no longer tells us anything useful, so we can
get rid of it:

2H +  J = 19

H + 2J = 14

Now we can use the same trick again:  If H + 2J = 14,
then H = 14 - 2J, so

2(14 - 2J) +  J = 19

(14 - 2J) + 2J = 14

Again, the final equation tells us nothing useful, so we can drop
it.  Now we have one equation, with one variable, which is good,
because that's the kind of thing we can solve:

2(14 - 2J) +  J = 19

28 - 4J + J = 19

28 - 3J = 19

28 - 19 = 3J

9 = 3J

3 = J

So Jacob scored 3 points, and you can use that information to
figure out what the other players scored.

But what if you _haven't_ had algebra yet?  Can you still solve a
problem like this?

Let's start with the fact that Harold and Isaac scored 19 points
together.  Here are the possibilities:

Harold   Isaac
-----    -----
19        0
18        1
17        2
16        3
15        4
14        5
13        6
12        7
11        8
10        9
9       10
8       11
7       12
6       13
5       14
4       15
3       16
2       17
1       18
0       19

Now, Isaac and Jacob scored a total of 14 points.  So we can add
another column to our table:

Harold   Isaac   Jacob
-----    -----   -----
19        0      14
18        1      13
17        2      12
16        3      11
15        4      10
14        5       9
13        6       8
12        7       7
11        8       6
10        9       5
9       10       4
8       11       3
7       12       2
6       13       1
5       14       0
4       15      -1
3       16      -2
2       17      -3
1       18      -4
0       19      -5

Now assuming that Jacob can't score negative points, we can
eliminate the final 6 rows of the table!

Now for our final piece of information:  Isaac scored as many
points as Harold and Jacob put together.  Let's add one more
column to the table:

Harold   Isaac   Jacob   Harold+Jacob
-----    -----   -----   ------------
19        0      14             33
18        1      13             31
17        2      12             29
16        3      11              ?
15        4      10              ?
14        5       9              ?
13        6       8              ?
12        7       7              ?
11        8       6              ?
10        9       5              ?
9       10       4              ?
8       11       3              ?
7       12       2              9
6       13       1              7

I'll leave the other values for you to fill in.  But notice that
in only _one_ of these rows will the values in the 2nd and 4th
columns be the same.  That row satisfies all the conditions in
the problem.

The thing about a problem like this is that it doesn't really
illustrate the power of algebra, because the table we had to
build to work the problem without algebra wasn't all that big.
But suppose that we had been working with numbers like 196 and
147 instead of numbers like 19 and 14?

The great thing about a solution using algebra is that it's about
the same size regardless of how large or small the numbers in the
problem are.

What that means in practice is that for small problems, it often
makes more sense to give trial-and-error a chance to work before
going to the trouble of setting up equations.  A big part of
becoming a good problem solver is learning to recognize when you
need steam roller to crush something, and when you can just use a

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 05/17/2002 at 14:46:32
From: M Hanlon
Subject: Math equation

Thanks so much for your help and time indoing this for me.  I can
now understand both ways of doing it.

M Hanlon
```
Associated Topics:
Middle School Equations

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