Section 3.4
Linear Equations 2
We consider here equations that have fractions in them, with the variable in the denominator, but leading to linear equations. As with other equations that have fractions, multiplying both sides of the equation by the least common denominator simplifies everything.
Examples:
a. \begin{align}
\frac{3}{x}+\frac{5}{2}&=\frac{7}{2x}\\
\Big(2x\Big)\cdot\frac{3}{x}+\frac{5}{2}&=\frac{7}{2x}\cdot\Big(2x\Big)&&\textit{Multiply both sides of the equation by}\\
&&&2x\textit{ the least common denominator}\\
6+5x&=7\\
5x&=1\\
x&=\frac{1}{5}
\end{align}
Algebra: Linear equations 4: Solving linear equations with variable expressions in the denominators of fractions
Exercieses
Solve for $x$.
\begin{array}{l l}
1. \;\;\frac{60}{x}-\frac{90}{2x}=\frac{1}{2}\;\;& 2. \;\;\frac{5}{x}+\frac{7}{3x}=1\;\; \\
3. \;\;\frac{2}{x}+\frac{4}{3x}=\frac{5x+10}{6x}\;\;& 4. \;\;\frac{3}{2x}+\frac{2}{x}=\frac{4x+9}{6x}\;\; \\
5. \;\;\frac{4}{x-2}+\frac{1}{3(x-2)}=\frac{13}{3}\;\;& 6. \;\;\frac{3}{2(x+1)}-\frac{1}{x+1}=\frac{1}{6}\;\; \\
7. \;\;\frac{2x}{x+2}+\frac{8}{2x+4}=2\;\;& 8. \;\;\frac{2}{x+2}-\frac{3}{2x+4}=\frac{1}{4}\;\; \\
9. \;\;\frac{3}{x}-\frac{1}{2x}=1\;\;& 10. \;\;\frac{3}{x+1}-\frac{1}{2}=\frac{x-5}{2x+2}\;\; \\
11. \;\;\frac{3}{x}+\frac{1}{2x}=\frac{3}{5}\;\;& 12. \;\;\frac{1}{3x}+\frac{1}{2x}=\frac{7}{5}\;\; \\
13. \;\;\frac{1}{6x-9}+\frac{1}{4x-6}=\frac{5}{6}\;\;& 14. \;\;5-\frac{a}{10x-2}=\frac{3a}{2}\;\; \\
15. \;\;\frac{a-1}{5x-1}-\frac{a}{10x-2}=\frac{1}{2}\;\;& 16. \;\;2+\frac{a}{x+3}=\frac{a}{2}\;\; \\
17. \;\;\frac{a}{3x+1}+\frac{2a+3}{6x+2}=1\;\;& 18. \;\;7-\frac{K}{3x-1}=\frac{K}{2}\;\; \\
19. \;\;\frac{2a}{x+2}+\frac{8}{2x+4}=2\;\;& 20. \;\;7-\frac{K}{2x+3}=\frac{5K}{4x+6}\;\;
\end{array}