Introduction - Linear Equations in One Variable - Chapter 2 - NCERT Class 8th Maths

Linear Equations in One Variable: Solving Equations and Word Problems

A linear equation in one variable is an equation of the form ax + b = c, where a, b, and c are constants and x is the unknown variable. Solving such an equation means finding the value of x that makes the equation true. This chapter in CBSE Class 8 Mathematics introduces the systematic process of solving linear equations by performing identical operations on both sides of the equation, and applies these skills to solve practical word problems involving numbers, ages, geometry, currencies, and everyday situations.

The fundamental principle of solving equations is that if you perform the same operation (addition, subtraction, multiplication, or division by a non-zero number) on both sides of an equation, the equation remains balanced. This is called the balancing method or transposition method. The process involves: (1) simplifying each side by combining like terms and removing brackets. (2) Transposing (moving) variable terms to one side and constant terms to the other side of the equation, changing the sign when crossing the equals sign. (3) Combining like terms on each side. (4) Dividing both sides by the coefficient of the variable to find its value. (5) Verifying the answer by substituting it back into the original equation. For example, to solve 3x + 5 = 17: subtract 5 from both sides → 3x = 12; divide both sides by 3 → x = 4. Verification: 3(4) + 5 = 12 + 5 = 17 ✓. When variables appear on both sides, move all variable terms to one side first: 5x − 3 = 3x + 7 → 5x − 3x = 7 + 3 → 2x = 10 → x = 5.

Equations reducible to linear form are those that initially have variables in the denominator or numerator but can be transformed into linear form by appropriate substitution. For example, x/(x+2) = 3/4 can be solved by cross-multiplying to get 4x = 3(x + 2), which simplifies to 4x = 3x + 6, giving x = 6. Word problems are the most practical application of linear equations. Common types include: (1) Number problems — "Five times a number decreased by 3 equals 22": 5x − 3 = 22 → x = 5. (2) Age problems — "Father is 3 times as old as his son; 5 years ago he was 5 times as old": set up two equations relating present ages and past ages. (3) Currency problems — "A bag contains 5 rupee and 10 rupee notes totalling 35 notes worth ₹250": let x be the number of 5-rupee notes, then (35 − x) is the number of 10-rupee notes, and 5x + 10(35 − x) = 250. (4) Perimeter/geometry problems — "The length of a rectangular park is 3 less than twice its width; its perimeter is 100 m": L = 2W − 3, 2(L + W) = 100. (5) Fraction problems — "The numerator of a fraction is 3 less than its denominator; adding 1 to both gives 1/2": (x−1)/(x+1) = 1/2. The key strategy is to identify the unknown, assign a variable, translate each condition into an equation, and solve systematically.

• A linear equation ax + b = c is solved by performing identical operations on both sides — addition, subtraction, multiplication, or division.
• Transposition: moving terms across the equals sign changes their sign; combine like terms after transposing.
• Equations with variables on both sides: collect all variable terms on one side and constants on the other.
• Word problems: identify the unknown, assign a variable, translate each condition into an equation, then solve and verify.
• Common types: number puzzles, age problems, currency problems, perimeter problems, and fraction problems.