The subjects of the preceding sections are introductory to what may be considered the peculiar province of algebra, the investigation of the values of unknown quantities, by means of equations.
An equation is a proposition, expressing in algebraic characters, the equality between one quantity or set of quantities and another, or between different expressions for the same quantity. Thus x + a = b + c, is an equation, in which the sum of x and a, is equal to the sum of b and c. The quantities on the two sides of the sign of equality, are sometimes called the members of the equation; the several terms on the left constituting the first member, and those on the right, the second member.
The object aimed at, in what is called the resolution or reduction of an equation, is to find the value of the unknown quantity. In the first statement of the conditions of a problem, the known and unknown quantities are frequently thrown promiscuously together. To find the value of that which is required, it is necessary to bring it to stand by itself, while all the others are on the opposite side of the equation. But in doing this, care must be taken not to destroy the equation, by rendering the two members unequal. Many changes may be made in the arrangement of the terms, without affecting the equality of the sides.
The reduction of an equation consists, then, in bringing the unknown quantity by itself, on one side, and all the known quantities on the other side, without destroying the equation.
To effect this, it is evident that one of the members must be as much increased or diminished as the other. If a quantity be added to one, and not to the other, the equality will be destroyed. But the members will remain equal;
If the same or equal quantities be added to each. Ax. 1.
If the same or equal quantities be subtracted from each. Ax. 2.
If each be multiplied by the same or equal quantities. Ax. 3.
If each be divided by the same or equal quantities. Ax. 4.
It may be further observed that, in general, if the unknown quantity is connected with others by addition, multiplication, division, &c. the reduction is made by a contrary process. If a known quantity is added to the unknown, the equation is reduced by subtraction. If one is multiplied by the other, the reduction is effected by division, &c. The reason of this will be seen, by attending to the several cases in the following articles. The known quantities may be expressed either by letters or figures. The unknown quantity is represented by one of the last letters of the alphabet, generally x, y, or z. The principal reductions to be considered in this section, are those which are effected by transposition, multiplication, and division. These ought to be made perfectly familiar, as one or more of them will be necessary, in the resolution of almost every equation.
169. In the equation
x - 7 = 9,
the number 7 being connected with the unknown quantity x by the sign -, the one is subtracted from the other. To reduce the equation by a contrary process, let 7 be added to both sides. It then becomes
x - 7 + 7 = 9 + 7.
The equality of the members is preserved, because one is as much increased as the other. (Axiom 1.) But on one side, we have -7 and +7. As these are equal, and have contrary signs, they balance each other, and may be cancelled. The equation will then be
x = 9 + 7.
Here the value of x is found. It is shown to be equal to 9 + 7, that is to 16. The equation is therefore reduced. The unknown quantity is on one side by itself, and all the known quantities on the other side.
In the same manner, if x - b = a
Adding b to both sides x - b + b = a + b
And cancelling (-b + b) x = a + b.
Here it will be seen that the last equation is the same as the first, except that b is on the opposite side, with a contrary sign.
Next suppose y + c = d.
Here c is added to the unknown quantity y. To reduce the equation by a contrary process, let c be subtracted from both sides, that is, let -c, be applied to both sides. We then have
y + c - c = d - c.
The equality of the members is not affected, because one is as much diminished as the other. When (+c - c) is cancelled, the equation is reduced, and is
y = d - c.
This is the same as y + c = d, except that c has been transposed, and has received a contrary sign. We hence obtain the following general rule:
170. When known quantities are connected with the unknown quantity by the sign + or -, the equation is reduced by transposing the known quantities to the other side, and prefixing the contrary sign.
This is called reducing an equation by addition or subtraction, because it is, in effect, adding or subtracting certain quantities, to or from, each of the members.
Ex. 1. Reduce the equation x + 3b - m = h - d
Transposing +3b, we have x - m = h - d - 3b
And transposing -m, x = h - d -3b + m.
171. When several terms on the same side of an equation are alike, they may be united in one, by the rules for reduction in addition.
Ex. 2. Reduce the equation x + 5b - 4h = 7b
Transposing 5b - 4h x = 7b - 5b + 4h
Uniting 7b - 5b in one term x = 2b + 4h.
172. The unknown quantity must also be transposed, whenever it is on both sides of the equation. It is not material on which side it is finally placed. For if x = 3, it is evident that 3 = x. It may be well, however, to bring it on that side, where it will have the affirmative sign, when the equation is reduced.
Ex. 3. Reduce the equation 2x + 2h = h + d + 3x
By transposition 2h - h - d = 3x - 2x
And h - d = x.
173. When the same term, with the same sign, is on opposite sides of the equation, instead of transposing, we may expunge it from each. For this is only subtracting the same qantity from equal quantities. (Ax. 2.)
Ex. 4. Reduce the equation x + 3h + d = b + 3h + 7d
Expunging 3h x + d = b + 7d
And x = b + 6d.
174. As all the terms of an equation may be transposed, or supposed to be transposed; and it is immaterial which member is written first; it is evident that, the signs of all the terms may be changed, without affecting the equality,
Thus, if we have x - b = d - a
Then by transposition -d + a = -x + b
Or, inverting the members -x + b = -d + a.
175. If all the terms on one side of an equation be transposed, each member will be equal to 0.
Thus, if x + b = d, then x + b - d = 0.
It is frequently convenient to reduce an equation to this form, in which the positive and negative terms balance each other. In the example just given, x + b is balanced by -d.
Ex. 5. Reduce a + 2x - 8 = b - 4 + x + a.
6. Reduce h + 30 + 7x = 8 - 6h + 6x - d + b.