Solving Equations with Variables on Both Sides
Applying balancing methods to isolate variables present on both sides of a linear equation.
Definition
These are linear equations that contain the same variable term on both sides of the equals sign. To solve them, you must use transposition to collect all variable terms on one side and all constant terms on the other side. Once grouped, simplify both sides and isolate the variable to find its value.
Key formula
ax + b = cx + d
Worked examples
Solve 3x - 2 = 2x + 1: Subtract 2x from both sides to get x - 2 = 1, then add 2 to get x = 3.
Solve 5p - 3 = 3p + 5: Transpose 3p to the left and -3 to the right to get 2p = 8, so p = 4.
Solve 2y + 5 = 7 - y: Transpose -y to get 3y = 2, yielding y = 2/3.
Common mistakes
- • Forgetting to change the sign of a term when transposing it across the equals sign.
- • Attempting to isolate constants and variables on opposite sides simultaneously without proper grouping.
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
Reducing Equations to Simpler Form
Linear Equations in One Variable