then a+a=6. But if a= any other number the sum would be different. Still as there are 2 a's we may say a+a=2a, and this will be true whatever number a stands for. If now we take a+a+a the result will be always 3a. So it is plain that if a represent any number, 2a will represent twice that number, 3a three times the number, 5a five times the number, and so on. The figure placed before the letter to show how many times it is taken, is called the numerical coefficient, or more shortly the coefficient.

Now suppose we wish to find the sum of 3a+5a. We know by arithmetic that the sum of 3 ones and 5 ones is 8 ones. Similarly the sum of 3 sevens and 5 sevens is 8 sevens, and so on for any other number. Therefore whenever 3 times a number is increased by 5 times that number, the result is 8 times the same number. Hence 3a+5a=8a, whatever number a represents. In like manner 4a+7a=11a, and 2a+4a +7a=13a. Here then we arrive at the first rule in the addition of like terms, i.e, of terms containing the same letter.

RULE 1.-Write down the sum of their coefficients and affix the common letter.

EXERCISE I.

x, 9x, 4x, 11x, 14x, 17x, 12x.

(1) Take any two of these terms and add them together. What is the numerical result in each case when x=5. (21 Examples.)

(2) Take any three of these terms and find the results as in example (1). (35 examples.)

(3) Take any four and obtain the results as before. (35 examples.)

When terms which are unlike are to be added together they cannot be collected into one term, but must

be written one after the other with the sign of addition between them. The actual addition can only be performed when we know what numerical values the symbols have. The sum of a and b is expressed thus a+b. In like manner the sum of 2x, 3y, 4z, is expressed thus 2x+3y+4z. But if a=1 and b=2, then a+b=3. And if x=1, y=2 and z=3, then 2x+3y+4z may be added together, and the sum would be 2×1+3×2+4×3=2+6+12=20.

If however some of the terms are like and others unlike the like terms may be placed under each other and added together by Rule 1. Thus, suppose we are to add together x+2y+3z, 5x+4y and 7x, there are three terms containing, and two containing y, which may be arranged under each other and added thus,

x+2y+3z
5x+4y
Тас

13x+6y+3z

The number of a's is 13; the y's amount to 6. The sum is therefore 13x+6y+3z. If x=1, y=2 and z=3, the numerical value of this result is 13×1+6×2 +3x3=13+12+9=34.

RULE 2.-When terms containing different letters are to be added together. Write in a column all the terms containing the same letter, find the sum of each column by Rule 1, and connect the results by the plus sign.

Add together,

EXERCISE II.

(1) 4x+3y+2z, 7x+y+8z, 3x+2y+9z.
(2) 13x+4y+z, 5x+9y+7z, 8x+11y+12z.
(3) 2a+3b+5z, 4a+7b+2z, 3a+9b+6z.

(4)*2xy+3yz+5xz,9xz+3xy+5yz, 8yz+4xz+3xy. (5) 13x+5z+8y, 7z+12y+3x, 10y+4x+12z. (6) 12yx + 10yz + 2xz, 13xy + xz+2yz, 5xz+2xy +yz.

(7) xy + y + z, x + yz + z, y+xz+x, yz+x+y, xz+z+y.

(8) xyz+yz+xy, xyz+2xz+2xy, xyz+3zy+4xy, xyz+4xz+5yz.

(9) 3x+5y+7z+xy, 4x+2y+3z+2xz and 5x+ 3y+2+3yz.

(10) 12% +10x + 15y, 2x + 3≈ +5y, 5x + 7y+ 8-, xy+2x+3yz.

*Find the numerical value of the next six results when x=1, y=2 and z=3.

SUBTRACTION (−).

When three is taken from five the result is two.. This operation we express in symbols thus 5-3=2. Similarly when 3a is to be taken from 5a we write 5a-3a, and the result is plainly 2a, for the same reason that three ones from five ones leaves 2 ones.

.. 5a-3a=2a. (1)

and 11a-6a=5a, and so on. But suppose we are required to take 7a from 5a, i.e., to find the result. of 5a-7a. From the minuend we cannot take away more a's than 5, whereas we wish to take away 7 a's. Thus, when we have taken away the greatest number possible, there will still remain 2 a's to be subtracted; and in order to show that 2 a's still remain unsubtracted, we write the answer -2a.

.. 5a-7a= − 2a (2)

means that the subtraction could not be completed because the quantity to be subtracted was greater by 2a than the quantity it was to be subtracted from.

Notice then that in both these cases:

(1) We prefix the minus sign to the quantity to be subtracted.

(2) We subtract the less coefficient from the greater. (3) We prefix to the answer the sign of the greater.

Next suppose that from -2a we were required to subtract 4a, i.e., to find the result of -2a-4a. Here we see that 2a is to be subtracted, and also 4a to be subtracted, making together 6a to be subtracted, i.e., -2a-4a= -6a. (3)

But now comes the most difficult case of all, which it is very important should be clearly understood. In all the three cases we have examined in writing down the example to be worked, the minus sign was prefixed to the quantity to be subtracted. But supposing that term already to have the minus sign before it, how then shall we proceed?

For instance, how shall we subtract - 2a from 8a? By looking at example (2) we see how we get such a quantity as -2a, viz., that it is the result of attempting to subtract a greater quantity from a less. Therefore for 2a we may write (5a-7a) placing them in Brackets to show that they are to be treated as one quantity. Our question now is, from 8a take (5a–7a), which we must write thus,

8a- (5a-7a).

Now if we take 5a from 8a the result is 3a. But this is greater than the required answer, because we were not to take away 5a but 7a less than that. We have therefore taken away 7a too much, and the answer will be 7a more than 3a i.e., 10a. Hence we get the true answer by first subtracting 5a and then adding 7a.

Thus 8a -(5a-7a) = 8a-5a+7a = 3a+7a=10a. Therefore Sa diminished by -2a is 10a. So that to obtain the true result the -2a must be changed to +2a, and added.

.. 8a -(—2a)=8a+2a=10a. (4)

Now let us collect all the four examples together, and then we shall see how to make a rule for the subtraction of like quantities.

(1) From 5a take 3a.
(2) From 5a take 7a.
-2a. This result

written 7a+5a.

This is written 5a-3a=2a. This is written 5a-7a= would be the same if it were

(3) From -2a take 4a. This is written -2a-4a

= -6a.

(4) From 8a take -2a.

= 10a.

=

This is written 8a+ 2a

If written under each other instead of in line, these examples would stand as follows:

:

Hence the following rule for the subtraction of like quantities.

RULE 3.-Write the terms under each other and change the sign of the quantity to be subtracted.

Then (1) If the signs are alike add the coefficients, prefix the common sign, and affix the common letter. (2) If the signs are different subtract the less coefficient from the greater, prefix the sign of the greater, and affix the common letter.

EXERCISE III.

(1) From +9a take away 7a, 9a, 11a, 22a, 3a, -7a, -9a, -12a, 20a, 40a.

(2) From - 10a take away 5a, 9a, 15a, — 12a, - 20a, -10a, -a, +8a, -7a.

RULE 4.-If there are several terms in the minuend and the subtrahend, the like terms are to be written under each other, and the subtraction performed according to Rule 3.