Skill: 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

$$\boxed{x=3}$$