Overview
An equation states that two expressions have the same value. Adding the same quantity to both sides or dividing both sides by the same nonzero quantity preserves that equality. Simplifying each side first often reveals the shortest route to isolating the unknown.
Consider this example: Solve 5x − 8 = 2x + 13. Subtract 2x and add 8 on both sides to get 3x = 21, so x = 7. Substitution gives 27 on both sides when x = 7. In contrast, x = 5 gives 17 on the left and 23 on the right, so it is not a solution.
The method also handles a different case: Solve 3(2x − 5) = 4x + 7. Distribute 3 to obtain 6x − 15 = 4x + 7. Then 2x = 22, giving x = 11. At x = 11, both original sides equal 51. The choice 4 gives 9 on the left and 23 on the right, so it fails the original equation.
For a further application, consider: Solve 2(x − 4) + 3(x + 1) = 4(x + 2). The left side is 5x − 5 and the right side is 4x + 8. Subtract 4x and add 5 to find x = 13. Substitution gives 60 on both sides.
Moving a term across the equals sign without reversing its sign is shorthand used incorrectly. A safer method writes the operation performed on both sides. After solving a new equation, substitute into the original version, including its parentheses. A solution must make both sides equal, not merely fit the last rewritten line.