Skill: Solve equations containing rational coefficients

Rational coefficients can be fractions or terminating decimals. To solve an equation with fractions, multiply every term by the least common denominator. This clears the fractions while keeping both sides equal.

Consider:

$$\frac{1}{2}x+\frac{3}{4}=\frac{5}{2}$$

Step 1: Find a common denominator.

The denominators are $2$ and $4$, so the least common denominator is $4$.

Step 2: Multiply every term by $4$.

$$4\left(\frac{1}{2}x\right)+4\left(\frac{3}{4}\right)=4\left(\frac{5}{2}\right)$$

Simplify:

$$2x+3=10$$

Step 3: Isolate the variable.

Subtract $3$ from both sides:

$$2x=7$$

Divide both sides by $2$:

$$x=\frac{7}{2}$$

Step 4: Check the solution.

Substitute $\frac{7}{2}$ for $x$ in the original equation:

$$\frac{1}{2}\left(\frac{7}{2}\right)+\frac{3}{4} =\frac{7}{4}+\frac{3}{4} =\frac{10}{4} =\frac{5}{2}$$

Both sides are equal, so the solution is

$$\boxed{x=\frac{7}{2}}$$