Solving equations with fractions involves using inverse operations to isolate the variable. One helpful strategy is to multiply every term by the least common denominator (LCD), which eliminates the fractions and makes the equation easier to solve. You can also add or subtract fractions using common denominators and multiply by reciprocals when appropriate. Always perform the same operation on both sides of the equation to keep it balanced.
Solve for x
Practice Problems
\(\textbf{1)}\) \(\displaystyle\frac{1}{3}x+5=8\) The answer is \(x=9\) \(\frac{1}{3}x+5=8\)
\(\frac{1}{3}x=3\)
\(3\cdot\frac{1}{3}x=3\cdot3\)
\(x=9\)
The answer is \(x=9\)
\(\frac{1}{3}x=3\)
\(3\cdot\frac{1}{3}x=3\cdot3\)
\(x=9\)
The answer is \(x=9\)
\(\textbf{2)}\) \(\displaystyle\frac{2}{5}x+\displaystyle\frac{2}{3}=\displaystyle\frac{8}{3}\) The answer is \(x=5\) \(\frac{2}{5}x+\frac{2}{3}=\frac{8}{3}\)
\(15\left(\frac{2}{5}x+\frac{2}{3}\right)=15\left(\frac{8}{3}\right)\)
\(6x+10=40\)
\(6x=30\)
\(x=5\)
The answer is \(x=5\)
\(15\left(\frac{2}{5}x+\frac{2}{3}\right)=15\left(\frac{8}{3}\right)\)
\(6x+10=40\)
\(6x=30\)
\(x=5\)
The answer is \(x=5\)
\(\textbf{3)}\) \(\displaystyle\frac{1}{3}x-\displaystyle\frac{9}{2}=\displaystyle\frac{9}{2}\) The answer is \(x=27\) \(\frac{1}{3}x-\frac{9}{2}=\frac{9}{2}\)
\(\frac{1}{3}x=\frac{9}{2}+\frac{9}{2}\)
\(\frac{1}{3}x=\frac{18}{2}\)
\(\frac{1}{3}x=9\)
\(x=9\cdot3\)
The answer is \(x=27\)
\(\frac{1}{3}x=\frac{9}{2}+\frac{9}{2}\)
\(\frac{1}{3}x=\frac{18}{2}\)
\(\frac{1}{3}x=9\)
\(x=9\cdot3\)
The answer is \(x=27\)
\(\textbf{4)}\) \(\displaystyle\frac{4}{7}+x=2\) The answer is \(x=\frac{10}{7}\) or \(1\frac{3}{7}\) \(\frac{4}{7}+x=2\)
\(x=2-\frac{4}{7}\)
\(x=\frac{14}{7}-\frac{4}{7}\)
\(x=\frac{10}{7}\)
The answer is \(x=\frac{10}{7}\) or \(1\frac{3}{7}\)
\(x=2-\frac{4}{7}\)
\(x=\frac{14}{7}-\frac{4}{7}\)
\(x=\frac{10}{7}\)
The answer is \(x=\frac{10}{7}\) or \(1\frac{3}{7}\)
\(\textbf{5)}\) \(\displaystyle\frac{3}{5}x-\frac{2}{3}=\frac{1}{5}\) The answer is \(x=\frac{13}{9}\) or \(1\frac{4}{9}\) \(\frac{3}{5}x-\frac{2}{3}=\frac{1}{5}\)
\(15\left(\frac{3}{5}x-\frac{2}{3}\right)=15\left(\frac{1}{5}\right)\)
\(9x-10=3\)
\(9x=13\)
\(x=\frac{13}{9}\)
The answer is \(x=\frac{13}{9}\) or \(1\frac{4}{9}\)
\(15\left(\frac{3}{5}x-\frac{2}{3}\right)=15\left(\frac{1}{5}\right)\)
\(9x-10=3\)
\(9x=13\)
\(x=\frac{13}{9}\)
The answer is \(x=\frac{13}{9}\) or \(1\frac{4}{9}\)
\(\textbf{6)}\) \(\displaystyle\frac{x}{3}-7=2\) The answer is \(x=27\) \(\frac{x}{3}-7=2\)
\(\frac{x}{3}=9\)
\(3\cdot\frac{x}{3}=9\cdot3\)
\(x=27\)
The answer is \(x=27\)
\(\frac{x}{3}=9\)
\(3\cdot\frac{x}{3}=9\cdot3\)
\(x=27\)
The answer is \(x=27\)
\(\textbf{7)}\) \(\displaystyle\frac{x}{5}+\displaystyle\frac{x}{6}=11\) The answer is \(x=30\) \(\frac{x}{5}+\frac{x}{6}=11\)
\(30\left(\frac{x}{5}+\frac{x}{6}\right)=30(11)\)
\(6x+5x=330\)
\(11x=330\)
\(x=30\)
The answer is \(x=30\)
\(30\left(\frac{x}{5}+\frac{x}{6}\right)=30(11)\)
\(6x+5x=330\)
\(11x=330\)
\(x=30\)
The answer is \(x=30\)
\(\textbf{8)}\) \(\displaystyle\frac{1}{4}x+\frac{1}{3}=\frac{5}{6}\) The answer is \(x=2\) \(\frac{1}{4}x+\frac{1}{3}=\frac{5}{6}\)
\(12\left(\frac{1}{4}x+\frac{1}{3}\right)=12\left(\frac{5}{6}\right)\)
\(3x+4=10\)
\(3x=6\)
\(x=2\)
The answer is \(x=2\)
\(12\left(\frac{1}{4}x+\frac{1}{3}\right)=12\left(\frac{5}{6}\right)\)
\(3x+4=10\)
\(3x=6\)
\(x=2\)
The answer is \(x=2\)
\(\textbf{9)}\) \(\displaystyle\frac{1}{5}x+\frac{1}{2}=\frac{7}{10}\) The answer is \(x=1\) \(\frac{1}{5}x+\frac{1}{2}=\frac{7}{10}\)
\(10\left(\frac{1}{5}x+\frac{1}{2}\right)=10\left(\frac{7}{10}\right)\)
\(2x+5=7\)
\(2x=2\)
\(x=1\)
The answer is \(x=1\)
\(10\left(\frac{1}{5}x+\frac{1}{2}\right)=10\left(\frac{7}{10}\right)\)
\(2x+5=7\)
\(2x=2\)
\(x=1\)
The answer is \(x=1\)
\(\textbf{10)}\) \(\displaystyle\frac{1}{5}x+\frac{2}{15}=\frac{7}{3}\) The answer is \(x=11\) \(\frac{1}{5}x+\frac{2}{15}=\frac{7}{3}\)
\(15\left(\frac{1}{5}x+\frac{2}{15}\right)=15\left(\frac{7}{3}\right)\)
\(3x+2=35\)
\(3x=33\)
\(x=11\)
The answer is \(x=11\)
\(15\left(\frac{1}{5}x+\frac{2}{15}\right)=15\left(\frac{7}{3}\right)\)
\(3x+2=35\)
\(3x=33\)
\(x=11\)
The answer is \(x=11\)
\(\textbf{11)}\) \(\displaystyle\frac{2}{5}x+\frac{2}{3}=\frac{8}{3}+2\) The answer is \(x=10\) \(\frac{2}{5}x+\frac{2}{3}=\frac{8}{3}+2\)
\(15\left(\frac{2}{5}x+\frac{2}{3}\right)=15\left(\frac{8}{3}+2\right)\)
\(6x+10=40+30\)
\(6x+10=70\)
\(6x=60\)
\(x=10\)
The answer is \(x=10\)
\(15\left(\frac{2}{5}x+\frac{2}{3}\right)=15\left(\frac{8}{3}+2\right)\)
\(6x+10=40+30\)
\(6x+10=70\)
\(6x=60\)
\(x=10\)
The answer is \(x=10\)
\(\textbf{12)}\) \(\frac{3}{4}x-\frac{1}{2}=4\) The answer is \(x=6\) \(\frac{3}{4}x-\frac{1}{2}=4\)
\(4\left(\frac{3}{4}x-\frac{1}{2}\right)=4(4)\)
\(3x-2=16\)
\(3x=18\)
\(x=6\)
The answer is \(x=6\)
\(4\left(\frac{3}{4}x-\frac{1}{2}\right)=4(4)\)
\(3x-2=16\)
\(3x=18\)
\(x=6\)
The answer is \(x=6\)
\(\textbf{13)}\) \(\frac{2}{3}x+\frac{1}{4}=\frac{9}{4}\) The answer is \(x=3\) \(\frac{2}{3}x+\frac{1}{4}=\frac{9}{4}\)
\(12\left(\frac{2}{3}x+\frac{1}{4}\right)=12\left(\frac{9}{4}\right)\)
\(8x+3=27\)
\(8x=24\)
\(x=3\)
The answer is \(x=3\)
\(12\left(\frac{2}{3}x+\frac{1}{4}\right)=12\left(\frac{9}{4}\right)\)
\(8x+3=27\)
\(8x=24\)
\(x=3\)
The answer is \(x=3\)
\(\textbf{14)}\) \(-\frac{2}{5}x+\frac{3}{4}=\frac{11}{4}\) The answer is \(x=-5\) \(-\frac{2}{5}x+\frac{3}{4}=\frac{11}{4}\)
\(20\left(-\frac{2}{5}x+\frac{3}{4}\right)=20\left(\frac{11}{4}\right)\)
\(-8x+15=55\)
\(-8x=40\)
\(x=-5\)
The answer is \(x=-5\)
\(20\left(-\frac{2}{5}x+\frac{3}{4}\right)=20\left(\frac{11}{4}\right)\)
\(-8x+15=55\)
\(-8x=40\)
\(x=-5\)
The answer is \(x=-5\)
\(\textbf{15)}\) \(\frac{x}{4}-\frac{x}{6}=2\) The answer is \(x=24\) \(\frac{x}{4}-\frac{x}{6}=2\)
\(12\left(\frac{x}{4}-\frac{x}{6}\right)=12(2)\)
\(3x-2x=24\)
\(x=24\)
The answer is \(x=24\)
\(12\left(\frac{x}{4}-\frac{x}{6}\right)=12(2)\)
\(3x-2x=24\)
\(x=24\)
The answer is \(x=24\)
\(\textbf{16)}\) \(\frac{x+3}{5}=4\) The answer is \(x=17\) \(\frac{x+3}{5}=4\)
\(5\left(\frac{x+3}{5}\right)=5(4)\)
\(x+3=20\)
\(x=20-3\)
\(x=17\)
The answer is \(x=17\)
\(5\left(\frac{x+3}{5}\right)=5(4)\)
\(x+3=20\)
\(x=20-3\)
\(x=17\)
The answer is \(x=17\)
\(\textbf{17)}\) \(\frac{2x-1}{3}+\frac{1}{2}=\frac{7}{2}\) The answer is \(x=5\) \(\frac{2x-1}{3}+\frac{1}{2}=\frac{7}{2}\)
\(6\left(\frac{2x-1}{3}+\frac{1}{2}\right)=6\left(\frac{7}{2}\right)\)
\(2(2x-1)+3=21\)
\(4x-2+3=21\)
\(4x+1=21\)
\(4x=20\)
\(x=5\)
The answer is \(x=5\)
\(6\left(\frac{2x-1}{3}+\frac{1}{2}\right)=6\left(\frac{7}{2}\right)\)
\(2(2x-1)+3=21\)
\(4x-2+3=21\)
\(4x+1=21\)
\(4x=20\)
\(x=5\)
The answer is \(x=5\)
\(\textbf{18)}\) \(\frac{3}{4}x+\frac{1}{2}=\frac{1}{4}x+5\) The answer is \(x=9\) \(\frac{3}{4}x+\frac{1}{2}=\frac{1}{4}x+5\)
\(4\left(\frac{3}{4}x+\frac{1}{2}\right)=4\left(\frac{1}{4}x+5\right)\)
\(3x+2=x+20\)
\(2x+2=20\)
\(2x=18\)
\(x=9\)
The answer is \(x=9\)
\(4\left(\frac{3}{4}x+\frac{1}{2}\right)=4\left(\frac{1}{4}x+5\right)\)
\(3x+2=x+20\)
\(2x+2=20\)
\(2x=18\)
\(x=9\)
The answer is \(x=9\)
\(\textbf{19)}\) \(\frac{x}{3}+\frac{x}{4}-\frac{1}{2}=3\) The answer is \(x=6\) \(\frac{x}{3}+\frac{x}{4}-\frac{1}{2}=3\)
\(12\left(\frac{x}{3}+\frac{x}{4}-\frac{1}{2}\right)=12(3)\)
\(4x+3x-6=36\)
\(7x-6=36\)
\(7x=42\)
\(x=6\)
The answer is \(x=6\)
\(12\left(\frac{x}{3}+\frac{x}{4}-\frac{1}{2}\right)=12(3)\)
\(4x+3x-6=36\)
\(7x-6=36\)
\(7x=42\)
\(x=6\)
The answer is \(x=6\)
\(\textbf{20)}\) \(\frac{x-2}{3}+\frac{x+1}{4}=7\) The answer is \(x=\frac{89}{7}\) or \(12\frac{5}{7}\) \(\frac{x-2}{3}+\frac{x+1}{4}=7\)
\(12\left(\frac{x-2}{3}+\frac{x+1}{4}\right)=12(7)\)
\(4(x-2)+3(x+1)=84\)
\(4x-8+3x+3=84\)
\(7x-5=84\)
\(7x=89\)
\(x=\frac{89}{7}\)
The answer is \(x=\frac{89}{7}\) or \(12\frac{5}{7}\)
\(12\left(\frac{x-2}{3}+\frac{x+1}{4}\right)=12(7)\)
\(4(x-2)+3(x+1)=84\)
\(4x-8+3x+3=84\)
\(7x-5=84\)
\(7x=89\)
\(x=\frac{89}{7}\)
The answer is \(x=\frac{89}{7}\) or \(12\frac{5}{7}\)
See Related Pages
\(\bullet\text{ Equation Calculator (Symbolab.com)}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Solving Equations by Addition and Subtraction}\)
\(\,\,\,\,\,\,\,\,x+3=4…\)
\(\bullet\text{ Solving Equations by Multiplication and Division}\)
\(\,\,\,\,\,\,\,\,8x=48…\)
\(\bullet\text{ Solving Multi-step Equations}\)
\(\,\,\,\,\,\,\,\,3x+2=14…\)
\(\bullet\text{ Solving Equations with Variables on Both Sides}\)
\(\,\,\,\,\,\,\,\,3x+5=7x-3…\)
\(\bullet\text{ Solving Equations with Decimals}\)
\(\,\,\,\,\,\,\,\,43.5+0.2x=51.1…\)
\(\bullet\text{ Solving Equations with Fractions}\)
\(\,\,\,\,\,\,\,\,\frac{2}{5}x+\frac{2}{3}=\frac{8}{3}…\)