CHAPTER X

FRACTIONAL EQUATIONS

83. Equations containing fractions are called Fractional Equations.

SOLUTION OF FRACTIONAL EQUATIONS

Since both sides of any equation may be multiplied by the same number without disturbing their equality, both sides of this equation may be multiplied by 12, the L. C. M. of the denominators. Each denominator being contained in 12, the denominators will be cancelled by this multiplication and the equation cleared of fractions.

Multiplying the equation by the L. C. M. of the denominators,

TO CLEAR AN EQUATION OF FRACTIONS

Factor the denominators and then multiply each term by the L. C. M. of the denominators.

In equations of this type, the work will be considerably shortened if, instead of clearing the equation of fractions, the fractions on each side of the equation are added.

On the left side, the L. C. D. is (x

1)(x 2) and on the right

side, (x − 3)(x − 4). Hence, adding the fractions, we have

In any equation containing monomial denominators as well as binomial or polynomial denominators, it is best to clear the equation of the monomial denominators by multiplying the equation by the L. C. M. of these monomials. In the above equation, the L. C. M. of 24, 8, and 4 is 24.

Combining like terms, the terms containing ≈ will disappear,

As each term of the equation is divisible by 2, divide the equation by 2 and thus reduce the size of the numbers.

36x+6
x − 1

As the remaining fraction is complex, simplify it by multiplying numerator and denominator by 3, which will not change the value of the fraction,

LITERAL EQUATIONS

85. Equations, in which some or all of the known numbers are expressed by letters, are called literal equations. It is usual to express known numbers by letters near the beginning of the alphabet.

Literal equations are solved by the same methods as numerical equations.

Transpose the terms containing the unknown number x to the left and the known terms to the right,

Note that collecting the coefficients of x is similar to collecting like terms. Thus, 6x - 4x =

(6-4)x

=

2x. In this case the like terms are added by adding their coefficients and, as these are numbers, they can actually be combined. If the coefficients are literal, this addition can only be indicated by connecting them by their proper signs and enclosing them in a parenthesis.

Add the fractions on each side. The L. C. M. on the left side

is (x — 2c)(x — 6c) and on the right

4c)(x

side (x
(x -8c)-(x

(x x + 2c

6c)

=

2c)(x-6c) (x

8c) 4c)

8c)

4c) (x

4c

4c)(x

x + 4c

8c)

8c)

As these equal fractions have equal numerators, their denominators must also be equal,