Overview
This lecture introduces how to solve equations that have variables on both sides, focusing on steps to isolate the variable and check solutions.
Steps for Solving Equations With Variables on Both Sides
- Goal: Get the variable (e.g., x, y) on one side of the equation.
- Choose which side to move the variable to, preferably avoiding negatives by moving the term with the smaller coefficient.
- Use inverse operations to eliminate the variable from one side (typically subtraction).
- Whatever operation is done on one side must be done on the other for balance.
- After having the variable on one side, further isolate it using inverse operations in the reverse order of operations.
- Check the solution by substituting it back into the original equation and verifying both sides are equal.
Example 1: Solving 5x - 4 = 2x + 11
- Subtract 2x from both sides: 5x - 2x - 4 = 2x - 2x + 11 → 3x - 4 = 11.
- Add 4 to both sides: 3x - 4 + 4 = 11 + 4 → 3x = 15.
- Divide both sides by 3: 3x / 3 = 15 / 3 → x = 5.
- Check by substituting x = 5 into the original equation; both sides equal 21.
Example 2: Solving 3y + 8 = 10y - 6
- Subtract 3y from both sides: 3y - 3y + 8 = 10y - 3y - 6 → 8 = 7y - 6.
- Add 6 to both sides: 8 + 6 = 7y - 6 + 6 → 14 = 7y.
- Divide both sides by 7: 14 / 7 = 7y / 7 → y = 2.
- Check by substituting y = 2 into the original equation; both sides equal 14.
Key Terms & Definitions
- Variable — A symbol, typically a letter, that represents an unknown value.
- Coefficient — The number in front of a variable in a term.
- Inverse Operation — An operation that undoes another (e.g., subtraction undoes addition).
- Isolate — To get the variable alone on one side of the equation.
Action Items / Next Steps
- Practice solving additional equations with variables on both sides.
- Review inverse operations and order of operations for future problems.