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EQUATIONS OF THE FIRST DEGREE. 125. An equation is an expression of equality between two algebraic quantities. Thus 3x=2ab is an equation denoting that three times the quantity x is equal to twice the product of the quantities a and b.
126. The first member of the equation is the quantity on the left side of the sign of equality, and the second member is the quantity on the right of the sign of equality. Thus, in the preceding equation, 3x is the first member, and 2ab the second member.
127. The two members of an equation are not only equal numerically, but must have the same essential sign. If, in the preceding equation, w represents a negative quantity, then the first member is essentially negative, and the second member must also be negative; that is, either a or b must represent a negative quantity.
128. Equations are usually composed of certain quantities which are known, and others which are unknown. The known quantities are represented either by numbers, or by the first letters of the alphabet; the unknown quantities are usually represented by the last letters of the alphabet.
129. A root of an equation is the value of the unknown quantity in the equation; or it is any value which, being substituted for the unknown quantity, will satisfy the equation. For example, in the equation
3x4=24-2, suppose x=7. Substituting 7 for x, the first member becomes 3x7-4; that is, 21 --4, or 17; and the second member becomes 24–7; that is, 17. Hence 7 is a root of the equation, because when substituted for a the two members are found to be equal.
130. A numerical equation is one in which all the known quantities are represented by figures; as, 2+4x2=3x+12.
131. A literal equation is one in which the known quantities are represented by letters, or by letters and figures. Thus 23 + ax +bo=m,
are literal equations.
132. The degree of an equation is denoted by the greatest number of unknown factors occurring in any term.
If the equation involves but one' unknown quantity, its degree is denoted by the exponent of the highest power of this quantity in any term.
If the equation involves more than one unknown quantity, its degree is denoted by the greatest sum of the exponents of the unknown quantities in any term.
Thus ax+b=cx+d is an equation of the first degree, and is sometimes called a simple equation.
4x2 — 2x=5-2% and 7xy—4x+y=40 are equations of the second degree, and are frequently called quadratic equations.
203 +ax?=25 and x2 + 3xy? +y=m are equations of the third degree, and are frequently called cubic equations.
So also we have equations of the fourth degree, sometimes called bi-quadratic equations; equations of the fifth degree, etc., up to the nth degree.
Thus och +ax–1=b is an equation of the nth degree.
133. To solve an equation is to find the value of the unknown quantity, or to find a number which, being substituted for the unknown quantity in the equation, renders the first member identical with the second.
The difficulty of solving equations depends upon their degree, and the number of unknown quantities they contain.
134. Axioms.—The various operations which we perform upon equations, in order to deduce the value of the unknown quantities, are founded upon the following principles, which are regarded as self-evident.
1. If to two equal quantities the same quantity be added, the sums will be equal.
2. If from two equal quantities the same quantity be subtracted, the remainders will be equal.
3. If two equal quantities be multiplied by the same quantity, the products will be equal.
4. If two equal quantities be divided by the same quantity, the quotients will be equal.
135. Transposition.—Transposition is the process of changing a term from one member of an equation to the other without destroying the equality of the members. Let it be required to solve the equation
+a=b. If from the two equal quantities ata and b we subtract the same quantity a, the remainders will be equal, according to the last article, and we shall have
x=b-a. Let it be required to solve the equation
X-a=b. If to the two equal quantities x-a and b the same quantity a be added, the sums will be equal, according to the last article, and we have x-ata=b+a,
136. Hence we perceive that we may transpose any term of an equation from one member of the equation to the other, provided we change its sign.
It is also evident that we may change the sign of every term of an equation without destroying the equality; for this is, in fact, the same thing as transposing every term in each member of the equation.
EXAMPLES. In the following examples, transpose the unknown terms to the first member and the known terms to the second member. 1. 5x+12=3x+18.
Ans. 5x – 3x=18-12. 2. 4x-7=21-3x.
Ans. 4x+3x=21+7. 3. 2-15= -7x+30.
Ans. 23+7x=30+15. 4. ax+bc=m-2x.
Ans. ax + 2x=m-bc. 5. 4ax-6+2c=3x—2ab-3mx.
Ans. 4ax-3x+3mx=b-2c-2ab. 6. 4ab-ax-2c=bx - 3m. Ans. ax+bx=4ab-2c+3m. 7. ab-cr-2mx=3ax-46.
Ans. 3ax+cx+2mx=ab +46.
137. To ciear an Equation of Fractions.—Let the equation be =3. If we multiply each of the equal quantities and b by the same quantity a, the products will be equal by Art. 134, and we shall have
Suppose the equation is to=m.
If we multiply each of the members of the equation by a, we shall have
b If we multiply each of the members of this equation by b, we shall have
bx+ ax=abm. Hence, to clear an equation of fractions, we have the following
Multiply each member of the equation by all the denominators.
X 3 1. Clear the equation
of fractions. 354
Ans. 20x— 12x=45.
3x 2x 3 2. Clear the equation
of fractions. 5 3
3. Clear the equation
2.0 32, 3
to=6 of fractions. 74 5
4. Clear the equation +*+=10 of fractions.
138. An equation may always be cleared of fractions by multiplying each member into all the denominators; but sometimes the same result may be attained by a less amount of multiplication. Thus, in the last example, the equation may be cleared of fractions by multiplying each term by 12 instead of 6 x4x2, and it is important to avoid all useless multiplication. In general, an equation may be cleared of fractions by multiplying each member by the least common multiple of all the denominators.
2x 3x 7 5. Clear the equation +
5 4 10 The least common multiple of all the denominators is 20. If we multiply each member of the equation by 20, we obtain
8x +15x=14. The operation is effected by dividing the least common multiple by each of the denominators, and then multiplying the corresponding numerator, dropping the denominator.
42 32 8 6. Clear the equation 7-1421
of fractions. 2-4
1 7. Clear the equation 3x
of fractions. 4
12 It should be remembered that when a fraction has the mi. nus sign before it, this indicates that the fraction is to be subtracted, and the signs of the terms derived from its numerator must be changed, Art. 118.
Ans. 362-3x+12=1. 8. Clear the equation
b ab Ans. a'-a-x-3ax+ 2ab=bx+ab?.