An equation is a mathematical statement that says two expressions are equal. For example, x + 5 = 9, 2y - 3 = 7 and p/4 = 6 are all equations. The letter in an equation is called the variable, because its value can vary or change. The expression on the left of the equals sign is the left hand side (LHS) and the expression on the right is the right hand side (RHS). An equation involving only one variable is called a linear equation in one variable when the variable appears only to the first power.
Solving an equation means finding the value of the variable that makes the equation true. This value is called the solution or the root of the equation. Equations are used to translate real-life situations into mathematics. For example, if a number added to 8 gives 20, we write x + 8 = 20 and solve it to find x = 12. Simple equations form the bridge between arithmetic and algebra, and every later chapter that involves unknown quantities depends on the skill of solving equations.
A word statement can be translated into an equation by choosing a variable for the unknown quantity and following the words. For example: - The sum of a number and 7 is 15 gives x + 7 = 15. - Twice a number minus 3 is 11 gives 2x - 3 = 11. - One third of a number is 9 gives x/3 = 9.
An equation is like a balance or a seesaw. Whatever operation we perform on one side must also be performed on the other side to keep the balance. If we add, subtract, multiply or divide one side by a number, we must do the same to the other side. This is called the balance method of solving equations.
In the trial and error method, we substitute values into the variable until the LHS equals the RHS. For example, for x + 5 = 12, trying x = 7 gives 7 + 5 = 12, which is correct, so x = 7 is the solution. This method works for simple equations but is slow for complex ones.
In the systematic method we perform the same operation on both sides. To solve x + 5 = 12, subtract 5 from both sides: x + 5 - 5 = 12 - 5, which gives x = 7. To solve 2x = 18, divide both sides by 2: 2x/2 = 18/2, which gives x = 9. To solve x/3 = 9, multiply both sides by 3: x = 27. To solve x - 4 = 10, add 4 to both sides: x = 14.
Transposing means moving a term from one side of the equation to the other while changing its sign. Addition becomes subtraction, subtraction becomes addition, multiplication becomes division and division becomes multiplication. For example, in x + 5 = 12, transpose 5 to the other side: x = 12 - 5 = 7. In 3x = 21, transpose 3: x = 21/3 = 7. Transposing is the quickest method and is simply a shortcut for applying the balance method.
To solve an equation with two operations, undo the operations in reverse order. For example, to solve 2x + 5 = 17, first subtract 5 from both sides to get 2x = 12, then divide both sides by 2 to get x = 6. To solve 3y - 4 = 20, first add 4 to both sides to get 3y = 24, then divide by 3 to get y = 8. Always undo addition or subtraction first, and then undo multiplication or division.
Sometimes the variable appears on both sides of the equation, like 3x + 2 = x + 10. Bring all the variable terms to one side and all the constant terms to the other side. Transpose x to the LHS and 2 to the RHS: 3x - x = 10 - 2, which gives 2x = 8, so x = 4.
Simple equations are used to solve many real-life puzzles and practical problems. For example: - Age problems: A father is 30 years older than his son. If the father is 42, the son's age x satisfies x + 30 = 42, so x = 12. - Number problems: If three times a number is increased by 8 gives 35, then 3x + 8 = 35, so 3x = 27 and x = 9. - Money problems: If a pen costs 4 rupees more than a pencil and the pen costs 14 rupees, the pencil's price x satisfies x + 4 = 14, so x = 10. - Geometry problems: The perimeter of a rectangle is given by 2(l + b); if perimeter is 48 and length is 15, we can solve 2(15 + b) = 48 to find b = 9.
| Method | Idea | Example |
|---|---|---|
| Trial and error | Substitute values until LHS = RHS | x + 5 = 12 gives x = 7 |
| Balance method | Same operation on both sides | x - 4 = 10, add 4 both sides, x = 14 |
| Transposing | Move term to other side, change sign | x + 5 = 12 becomes x = 12 - 5 = 7 |
| Reverse operations | Undo + then x, in reverse order | 2x + 5 = 17 gives x = 6 |
| Equation Type | Steps | Solution |
|---|---|---|
| x + a = b | Transpose a | x = b - a |
| x - a = b | Transpose -a | x = b + a |
| ax = b | Transpose a (divide) | x = b/a |
| x/a = b | Transpose a (multiply) | x = a x b |
| ax + b = c | Transpose b, then divide by a | x = (c - b)/a |
| ax + b = cx + d | Group variable terms, then solve | x = (d - b)/(a - c) |
Simple equations translate everyday situations into mathematical form and provide a systematic way to find unknown values. Whether we use trial and error, the balance method, or transposing, the core idea is the same: keep the two sides balanced while isolating the variable. Once an equation is solved, verification by substitution confirms the correctness of the answer. Mastery of simple equations prepares students for linear equations in two variables, algebraic expressions and word problems in higher classes, and it develops the logical step-by-step thinking that is the essence of algebra.