Chapter № 𝐼𝐼; Negative Quantities, Addition of Like Terms.

CHAPTER 𝐼𝐼. 

Negative Quantities. Addition of Like Terms.

𝟏𝟗. In his arithmetical work the student has been accustomed to deal with numerical quantities connected by the signs + and — ; and in finding the value of an expression such as 𝟏 𝟑/𝟒 + 𝟕 𝟐/𝟑 — 𝟑 𝟏/𝟖 + 𝟔 ― 𝟒 𝟏/𝟓 he understands that the quantities to which the sign-f- is prefixed are additive, and those to which the sign — is prefixed are subtractive, while the first quantity, 𝟏 𝟑/𝟒, to which no sign is prefixed, is counted among the additive terms. The same notions prevail in Algebra; thus in using the expression 𝟕𝑎 + 𝟑𝑏 — 𝟒𝑐 — 𝟐𝑑 we understand the symbols 𝟕𝑎 and 𝟑𝑏 to be additive, while 𝟒𝑐 and 𝟐𝑑 are subtractive.

𝟐𝟎. But in Arithmetic the sum of the additive terms is always greater than the sum of the subtractive terms; and if the reverse were the case, the result would have no arithmetical meaning. In Algebra, however, not only may the sum of the subtractive terms exceed that of the additive, but a subtractive term may stand alone, and yet have a meaning quite intelligible. Hence all algebraic quantities may be divided into positive quantities and negative quantities, according as they are expressed with the sign + or the sign — ; and this is quite irrespective of any actual process of addition and subtraction. 

This idea may be made clearer by one or two simple illustrations. 

(𝒊) Suppose a man were to gain $𝟏𝟎𝟎 and then lose $𝟕𝟎, his total gain would be $𝟑𝟎. But if he first gains $𝟕𝟎 and then loses $𝟏𝟎𝟎 the result of his trading is a loss of $𝟑𝟎.

The corresponding algebraic statements would be 

                        $𝟏𝟎𝟎 ― $𝟕𝟎 = +   $𝟑𝟎, 

                        $𝟕𝟎 ― $𝟏𝟎𝟎 = ― $𝟑𝟎, 

and the negative quantity in the second case is interpreted as a debt, that is, a sum of money opposite in character to the positive quantity, or gain, in the first case; in fact it may be said to possess a subtractive quality which would produce its effect on other transactions, or perhaps wholly counterbalance a sum gained.

(𝒊𝒊) Suppose a man starting from a given point were to walk along a straight road 𝟏𝟎𝟎 yards forwards and then 𝟕𝟎 yards backwards, his distance from the starting-point would be 30 yards. But if he first walks 𝟕𝟎 yards forwards and then 𝟏𝟎𝟎 yards backwards his distance from the startingpoint would be 𝟑𝟎 yards, but on the opposite side of it. As before we have 

                        𝟏𝟎𝟎 yards — 𝟕𝟎 yards = + 𝟑𝟎 yards, 

                      𝟕𝟎 yards — 𝟏𝟎𝟎 yards = — 𝟑𝟎 yards. In each of these cases the man's absolute distance from the starting-point is the same; but by taking the positive and negative signs into account, we see that — 𝟑𝟎 is a distance from the starting-point equal in magnitude but opposite in direction to the distance represented by + 𝟑𝟎. Thus the negative sign may here be taken as indicating a reversal of direction. Many other illustrations might be chosen; but it will be sufficient here to remind the student that a subtractive quantity is always opposite in character to an additive quantity of equal absolute value. 

Note. Absolute value is the value taken independently of the signs + and —. It is denoted by vertical bars, e.g., |𝒙|, where the quantity inside the bars is taken without regard to its sign.

𝟐𝟏. Definition. When any number of quantities are connected by the signs + and —, the resulting expression is called their algebraic sum. Thus 𝟏𝟏𝑎 — 𝟐𝟕𝑎 + 𝟏𝟑𝑏 — 𝟓𝑏 is an algebraic sum. This expression, however, is not, as will be shown, in its simplest form.

𝟐𝟐. Addition is the process of finding in simplest form the algebraic sum of any number of quantities.

𝟐𝟑. Like terms, or similar terms, do not differ, or differ only in their numerical coefficients. Other terms are called unlike, or dissimilar. Thus 𝟑𝑎, 𝟕𝑎; 𝟓𝑎²𝑏, 𝟐𝑎²𝑏; 𝟑𝑎³𝑏², — 𝟒𝑎³𝑏² are pairs of like terms; and 𝟒𝑎, 𝟑𝑏; 𝟕𝑎², 𝟗𝑎²𝑏 are pairs of unlike terms.

ADDITION OF LIKE TERMS. 

Rule 𝐼. The sum of a number of like terms is a like term. 

Rule 𝐼𝐼. If all the terms are positive, add the coefficients.

Ex. Find the value of 𝟖𝑎 + 𝟓𝑎. Here we have to increase 𝟖 like things by 𝟓 like things of the same kind, and the aggregate is 𝟏𝟑 of such things; 

for instance,                   𝟖 lbs. + 𝟓 lbs. = 𝟏𝟑 lbs. 

Hence also,                     𝟖𝑎 + 𝟓𝑎 = 𝟏𝟑𝑎

Similarly,                         𝟖𝑎 + 𝟓𝑎 + 𝑎 + 𝟐𝑎 + 𝟔𝑎 = 𝟐𝟐𝑎.

Rule 𝐼𝐼𝐼. If all the terms are negative, add the coefficients numerically and prefix the minus sign to the sum. 

Ex. To find the sum of — 𝟑𝒙, — 𝟓𝒙, — 𝟕𝒙, — 𝒙. Here the word sum indicates the aggregate of 𝟒 subtractive quantities of like character. In other words, we have to take away successively 𝟑, 𝟓, 𝟕, 𝟏 like things, and the result is the same as taking away 𝟑 + 𝟓 + 𝟕 + 𝟏 such things in the aggregate. 

Thus the sum of — 𝟑𝒙, — 𝟓𝒙, — 𝟕𝒙, — 𝒙, is — 𝟏𝟔𝒙.

Rule 𝐼𝑉. If the terms are not all of the same sign, add together separately the coefficients of cdl the positive terms and the coefficients of all the negative terms; the difference of these two results, preceded by the sign of the greater, will give the coefficient of the sum required.

Ex. 𝟏. The sum of 𝟏𝟕𝒙 and — 𝟖𝒙 is 𝟗𝒙, for the difference of 𝟏𝟕 and 𝟖 is 𝟗, and the greater is positive. 

Ex. 𝟐. To find the sum of 𝟖𝑎, — 𝟗𝑎, — 𝑎, 𝟑𝑎, 𝟒𝑎, —𝟏𝟏𝑎, 𝑎. 

The sum of the coefficients of the positive terms is 𝟏𝟔. 

The sum of the coefficients of the negative terms is 𝟐𝟏.  

The difference of these is 𝟓, and the sign of the gieater is negative ; lience the required sum is — 𝟓𝑎.

We need not, however, adhere strictly to this rule, for the terms may be added or subtracted in the order we find most The sum convenient.

This process is called collecting terms. 

Ex. 𝟑. Find the sum of 𝟐/𝟑 𝑎, 𝟑 𝑎, — 𝟏/𝟔 𝑎, ― 𝟐𝑎.

The sum               = 𝑎 (𝟐/𝟑 + 𝟑 — 𝟏/𝟔 ― 𝟐) 

                             = 𝑎 (𝟐/𝟑 + 𝟑 ― 𝟐 — 𝟏/𝟔) = 𝑎 (𝟐/𝟑 —                                   𝟏/𝟔 + 𝟏) 

                              =  𝑎 (𝟒/𝟔 — 𝟏/𝟔 + 𝟏) 

                              =  𝑎 (𝟑/𝟔 + 𝟏) 

                              =  𝑎 (𝟏/𝟐 + 𝟏) 

                              = 𝑎 (𝟑/𝟐) 

                              = 𝟑/𝟐 𝑎.

                              = 𝟑 𝟐/𝟑 𝑎 — 𝟐 𝟏/𝟔 𝑎 

                              = 𝟏 𝟏/𝟐 

                              = 𝟑/𝟐 𝑎. 

Note. The sum of two quantities numerically equal but with opposite signs is zero. The sum of 𝟓𝑎 and — 𝟓𝑎 is 𝟎.

EXAMPLES II. 

Find the sum of 

𝟏. 𝟓𝑎, 𝟕𝑎, 𝟏𝟏𝑎, 𝑎, 𝟐𝟑𝑎. 

𝟐. 𝟒𝒙, 𝒙, 𝟑𝒙, 𝟕𝒙, 𝟗𝒙. 

𝟑. 𝟕𝑏, 𝟏𝟎𝑏, 𝟏𝟏𝑏, 𝟗𝑏, 𝟐𝑏. 

𝟒. 𝟔𝑐, 𝟖𝑐, 𝟐𝑐, 𝟏𝟓𝑐, 𝟏𝟗𝑐, 𝟏𝟎𝟎𝑐, 𝑐. 

𝟓. ― 𝟑𝒙, ― 𝟓𝒙, ― 𝟏𝟏𝒙, ― 𝟕𝒙. 

𝟔. ― 𝟓𝑏, ― 𝟔𝑏, ― 𝟏𝟏𝑏, ― 𝟏𝟖𝑏. 

𝟕. ― 𝟑𝒚, ― 𝟕𝒚, ― 𝒚, ― 𝟐𝒚, ― 𝟒𝒚. 

𝟖. ― 𝑐, ― 𝟐𝑐, ― 𝟓𝟎𝑐, ― 𝟏𝟑𝑐, 

𝟗. ― 𝟏𝟏𝑏, ― 𝟓𝑏, ― 𝟑𝑏, 𝑏. 

𝟏𝟎. 𝟓𝒙, ― 𝒙, ― 𝟑𝒙, 𝟐𝒙, ― 𝒙. 

𝟏𝟏. 𝟐𝟔𝒚, ― 𝟏𝟏𝒚,  𝒚, ― 𝟏𝟓𝒚, ― 𝟑𝒚, 𝟐𝒚. 

𝟏𝟐.𝟓 𝑓, ― 𝟗 𝑓, ― 𝟑𝑓, 𝟐𝟏𝑓, ― 𝟑𝟎𝑓. 

𝟏𝟑. 𝟐𝒔, ― 𝟑𝒔, 𝒔, ― 𝒔, ― 𝟓𝒔, 𝟓𝒔. 

𝟏𝟒. 𝟕𝒚, ― 𝟏𝟏𝒚, 𝟏𝟔𝒚, ― 𝟑𝒚, ― 𝟐𝒚. 

𝟏𝟓. 𝟓𝒙, ― 𝟕𝒙, ― 𝟐𝒙, 𝟕𝒙, 𝟐𝒙, ―𝟓𝒙. 

𝟏𝟔. 𝟕𝑎𝑏, ― 𝟑𝑎𝑏, ― 𝟓𝑎𝑏, 𝟐𝑎𝑏, 𝑎𝑏. 

Find the value of

𝟏𝟕. ― 𝟗𝒙² + 𝟏𝟏𝒙² + 𝟑𝒙² ― 𝟒𝒙². 

𝟏𝟖. 𝟑𝑎²𝒙 + 𝟏𝟖𝑎²𝒙 + 𝑎²𝒙. 

𝟏𝟗. 𝟑𝑎³ ― 𝟕𝑎³ ― 𝟖𝑎³ + 𝟐𝑎³ ― 𝟏𝟏𝑎³. 

𝟐𝟎. 𝟒𝒙³ ― 𝟓𝒙³ ― 𝟖𝒙³ ― 𝟕𝒙³. 

𝟐𝟏. 𝟒𝑎²𝑏² ― 𝑎²𝑏² ― 𝟕𝑎²𝑏² + 𝟓𝑎²𝑏² ― 𝑎²𝑏². 

𝟐𝟐. ― 𝟗𝒙⁴ ― 𝟒𝒙⁴ ― 𝟏𝟐𝒙⁴ + 𝟏𝟑𝒙⁴ ― 𝟕𝒙⁴. 

𝟐𝟑. 𝟕𝑎𝑏𝑐𝑑 ― 𝟏𝟏𝑎𝑏𝑐𝑑 ― 𝟒𝟏𝑎𝑏𝑐𝑑 + 𝟐𝑎𝑏𝑐𝑑. 

𝟐𝟒. 𝟏/𝟐 𝒙 ― 𝟏/𝟑𝒙 + 𝒙 + 𝟐/𝟑𝒙. 

𝟐𝟓. 𝟑/𝟐 𝑎 + 𝟑/𝟓 𝑎 ― 𝟏/𝟐 𝑎. 

𝟐𝟔. ― 𝟓𝑏 + 𝟏/𝟒 𝑏 ― 𝟑/𝟐 𝑏 + 𝟐𝑏 ― 𝟏/𝟐 𝑏 + 𝟕/𝟒 𝑏. 

𝟐𝟕. ― 𝟓/𝟑 𝒙² ― 𝟐𝒙² ― 𝟐/𝟑 𝒙² + 𝒙² + 𝟏/𝟐 𝒙² + 𝟏𝟏/𝟔 𝒙². 

𝟐𝟖. ― 𝑎𝑏 ―𝟏/𝟐 𝑎𝑏 ― 𝟏/𝟑 𝑎𝑏 ―𝟏/𝟒 𝑎𝑏 ― 𝟏/𝟔𝑎𝑏 + 𝑎𝑏 + 𝟓/𝟏𝟐 𝑎𝑏. 

𝟐𝟗. 𝟐/𝟑 𝒙 ― 𝟑/𝟒 𝒙 + 𝟓/𝟔 𝒙 ― 𝟐𝒙 + 𝟏𝟏/𝟔 𝒙 ― 𝟏/𝟑 𝒙 + 𝒙. 

𝟑𝟎. ―𝟓/𝟑 𝒙² ― 𝟑/𝟒 𝒙² ― 𝟒/𝟑 𝒙² ― 𝟏/𝟒 𝒙² ― 𝒙². 

By Sohail Tahir.