Fractional Equations

Fractional Equations.

139. To Reduce Equations containing Fractions.

1. Solve ${\displaystyle \frac{x}{3}}-{\displaystyle \frac{x-1}{11}}=x-9$.

Multiply by $33$, the L. C. M. of the denominators.

lint Then, 11x - 3x + 3 = 33x - 297. lint Transpose, 11x - 3x - 33x = - 297 - 3. lint Combine, - 25x = - 300. lint Divide by $-25$, x = 12.

Since the minus sign precedes the second fraction, in removing the denominator, the $+$ (understood) before $x$, the first term of the numerator, is changed to $-$, and the $-$ before $1$, the second term of the numerator, is changed to $+$.

clear an equation of fractions, therefore,

Multiply each term by the L. C. M. of the denominators.

If a fraction is preceded by a minus sign, the sign of every term of the numerator must be changed when the denominator is removed.

2. Solve ${\displaystyle \frac{x+1}{4}}-\frac{1}{5}(x-1)=1$.

Multiply by $20$, the L. C. D.

3. Solve $7x-\frac{(2x-3)(3x-5)}{5}=\frac{153}{10}-\frac{(4x-5)(3x-1)}{10}.$

Multiply by $10$, the L. C. D., and we have $70x-2(2x-3)(3x-5)=153-(4x-5)(3x-1).$

Find the products of $(2x-3)(3x-5)$ and $(4x-5)(3x-1)$. $70x-2(6x^2-19x+15)=153-(12x^2-19x+5).$

Remove the parentheses, $70x-12x^2+38x-30=153-12x^2+19x-5.$

Cancel the $-12x^2$ on each side and transpose,

70x + 38x - 19x = 153 + 30 - 5. lint Combine, 89x = 178. lint Divide by $89$, x = 2.

4. Solve ${\displaystyle \frac{2x+1}{2x-1}}-{\displaystyle \frac{2x-1}{2x+1}}={\displaystyle \frac{8}{4x^2-1}}$.

Multiply by the L. C. D., and we have,

5. Solve ${\displaystyle \frac{4}{x+1}}-{\displaystyle \frac{x+1}{x-1}}+{\displaystyle \frac{x^2-3}{x^2-1}}=0$.