Reducing Equations to Simpler Form
Simplifying complex linear equations into standard forms before solving them.
Definition
Sometimes linear equations are presented in complex forms involving parentheses, fractions, or decimals. Reducing them involves clearing fractions by multiplying by the LCM of the denominators, expanding brackets using distributive laws, and clearing decimals. After these steps, the equation transforms into a standard linear form that is easy to solve.
Key formula
(x + a)/b = (cx + d)/e
Worked examples
Solve (x - 5)/3 = (x - 3)/5: Cross-multiply to get 5(x - 5) = 3(x - 3), expand to 5x - 25 = 3x - 9, which gives 2x = 16 or x = 8.
Solve 3(t - 3) = 5(2t + 1): Expand to 3t - 9 = 10t + 5, transpose to get -7t = 14, so t = -2.
Solve x/2 - 1/5 = x/3 + 1/4: Multiply the entire equation by the LCM of denominators (60) to eliminate fractions.
Common mistakes
- • Multiplying only the numerator and forgetting to multiply the denominator when clearing fractions.
- • Applying the distributive property to only the first term inside the parentheses instead of all terms.
Introduction to Rational Numbers
Rational Numbers
Properties of Rational Numbers
Rational Numbers
Representation on Number Line
Rational Numbers
Rational Numbers Between Two Rational Numbers
Rational Numbers
Algebraic Expressions and Equations
Linear Equations in One Variable
Solving Equations with Variables on Both Sides
Linear Equations in One Variable