Solve equations containing fractions

This skill involves interpreting and solving linear equations in which constants, coefficients, or both are rational numbers, including fractions and mixed numbers, by maintaining equivalence through inverse operations or by multiplying every term by a nonzero least common denominator. The reasoning includes distributing multiplication correctly, combining like terms, and checking the solution in the original equation; equations with variables in denominators, complex rational expressions, and other more advanced rational-equation cases are not included.

Detailed Explanation: Solve equations containing fractions

To solve an equation containing fractions, you can clear the fractions by multiplying every term on both sides by the least common denominator (LCD). This keeps the equation equivalent while making it easier to solve.

Consider:

\frac{2}{3}x+\frac14=\frac16x+\frac74

1. Find the LCD

The denominators are $3$ , $4$ , $6$ , and $4$ . Their least common denominator is $12$ .

Multiply every term by $12$ :

12\left(\frac23x\right)+12\left(\frac14\right) =12\left(\frac16x\right)+12\left(\frac74\right)

2. Simplify each term

8x+3=2x+21

The fractions are gone.

3. Collect the variable terms

Subtract $2x$ from both sides:

6x+3=21

Subtract $3$ from both sides:

6x=18

Divide both sides by $6$ :

x=3

4. Check the solution

Substitute $x=3$ into the original equation:

\frac23(3)+\frac14=\frac16(3)+\frac74

2+\frac14=\frac12+\frac74

\frac94=\frac24+\frac74=\frac94

Both sides are equal, so the solution is

\boxed{x=3}

Learn by doing: Solve equations containing fractions

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Linear Equation - One Variable, Three Terms

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