SIMPLE EQUATIONS.

141. An EQUATION is an expression of equality between two quantities. Thus,

x+4= 16

is an equation, expressing the equality of the quantities x+4 and 16.

142. The FIRST MEMBER of an equation is the quantity on the left of the sign of equality.

The SECOND MEMBER is the quantity on the right of the same sign.

The sides of an equation are its two members.

143. An IDENTICAL EQUATION is one in which the two members are the same algebraic expression, or become the same on performing the operations indicated; as,

х y = xy, or 2a2bc 2 (a+be).

144. A NUMERICAL EQUATION is one in which all the known quantities are expressed by figures; as,

145. A LITERAL EQUATION is one in which some or all the known quantities are expressed by letters; as,

2x + a = x2 10.

146. The DEGREE of an equation containing but one unknown quantity is denoted by the exponent of the highest power of the unknown quantity in the equation. Thus,

An equation of the first degree is one having no higher power of the unknown quantity than its first power; as,

An equation of the second degree is one in which the highest power of the unknown quantity is the second power, or square; as,

In like manner, we have equations of the third degree, fourth degree, nth degree.

When an equation contains more than one unknown quantity, its degree is determined by the greatest sum of the exponents of the unknown quantities, in any term Thus,

xxy=25 is an equation of the second degree. x2- y2 za b3 is an equation of the third degree.

147. A SIMPLE EQUATION is an equation of the first degree.

148. The Roor of an equation is the value of its unknown quantity.

The root is verified, or the equation satisfied, when, the root being substituted for its symbol, the equation becomes identical.

TRANSFORMATION OF EQUATIONS.

149. The TRANSFORMATION of an equation is the process of changing its form without destroying the equality.

150. The operations required in the transformation are based upon the general principle deduced directly from the axioms (Art. 46):—

If the same operations are performed upon equal quantities, the results will be equal; hence,

Both members of an equation may be increased, diminished, multiplied, or divided by the same quantity, without destroying the equality.

CASE I.

151. To transpose terms of an equation.

TRANSPOSITION is the process of changing terms from one member of an equation to the other, without destroying the equality.

or,

1. Let it be required to transpose

x— a b.

a in

Adding a to each member (Art. 150); then

x = a + a=b+

x = b+a,

Wherea has been transposed with the sign changed. 2. Let it be required to transpose a in

x + a = b.

Subtracting a from each member (Art. 150); then

Where a has been transposed with the sign changed.

RULE.

Any term may be transposed from one member of an equation to the other, provided its sign be changed.

If the same term appear in both members of an equation, affected with the same sign, it may be suppressed.

It also follows, since every term may be transposed, that the SIGNS of all the terms of an equation may be changed, without destroying the equality.

Transpose the unknown terms to the first member, and the known terms to the second, in the following

Ans. 7xcx 2a-acad.

7. bc + a2x-mn2 = bx+ad-5.

Ans. axbx ad-5-bcm n2.

CASE II.

152. To clear an equation of fractions.

1. Let it be required to clear of fractions

Since multiplying a fraction by a quantity equal to its denominator will cancel the denominator, if both members of the equation be multiplied by ac and by be, or by ac be, the fractions will be removed, without destroying the equality (Art. 150).

But, multiplying by any multiple of the denominators will cause them to disappear; hence, multiplying by their least common multiple, abc, as the least possible multiple that will transform the fractions to the form of entire quantities, we have

bx+ax = abcd.

Hence the

RULE.

Multiply each term of the equation by all the denominators, or by the least common multiple of the denominators.

When a fraction is preceded by since it is to be

subtracted, on removing its denominator, all the signs of its numerator must be changed (Art. 125).

Negative exponents are removed from an equation in the same way as fractions.

Clear the equations of fractions in the following

153. The SOLUTION OF AN EQUATION is the process of finding the value, or values, of the unknown quantity, or the roots of the equation.

154. To solve an equation, such transformations are required as shall make the unknown quantity stand alone as one member of the equation, and all the known quantities as the other member.