Equations with Fractions and Decimals

Objective: Solve a linear equation containing a fraction.

Prerequisite skills: equation balance and fraction arithmetic. Multiplying both sides by a nonzero denominator removes that denominator while preserving equality. Keep the numerator grouped until after multiplication. Common error: multiplying only x and leaving the added constant unchanged. Restrictions and interpretation: Every denominator is nonzero; clearing fractions multiplies every term, not just fractions.

Objective: Use a least common denominator on both sides.

For numerical denominators, multiply every term by their least common multiple. This turns the equation into an integer-coefficient equation. Simplify before collecting variables. Common error: forgetting to multiply the term that has no denominator. Restrictions and interpretation: Every denominator is nonzero; clearing fractions multiplies every term, not just fractions.

Objective: Convert decimal coefficients to integers before solving.

Multiplying by a power of ten removes finite decimal places. Apply it uniformly, then solve. A terminating decimal is exact unless the problem identifies it as an approximation. Common error: shifting the decimal on only one side. Restrictions and interpretation: Every denominator is nonzero; clearing fractions multiplies every term, not just fractions. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.