worldofcalcsEquation Solver
Consistent with a unique solution: the lines intersect at one point, since a₁/a₂ ≠ b₁/b₂.
What equations can this solve?
Three kinds from the NCERT syllabus. Linear equations in one variable, like 2x + 5 = 11 (Class 8). Pairs of linear equations in two variables, like x + y = 14 and x − y = 4 (Class 10), including whether they have one solution, none or infinitely many. And quadratic equations, like 2x² − 5x + 3 = 0, with the discriminant and the nature of the roots (Class 10). Answers are shown as exact fractions where possible.
Key formulas
The middle formulas are the cross-multiplication method for a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The last is the quadratic formula; its discriminant D = b² − 4ac decides the nature of the roots.
Nature of solutions
- A pair of linear equations has a unique solution if a₁/a₂ ≠ b₁/b₂, no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and infinitely many if all three ratios are equal.
- Graphically, these are intersecting, parallel and coincident lines.
- A quadratic has two distinct real roots when D > 0, two equal roots when D = 0 and no real roots when D < 0.
- The sum of a quadratic's roots is −b/a and their product is c/a: for 2x² − 5x + 3 = 0, 3/2 + 1 = 5/2 and 3/2 × 1 = 3/2.
- Always check a solution by substituting it back into the original equation.
Worked examples
- 2x + 5 = 11 ⇒ 2x = 6 ⇒ x = 3
- x + y = 14 and x − y = 4: a₁b₂ − a₂b₁ = (1)(−1) − (1)(1) = −2
- x = (14 × (−1) − 4 × 1) ÷ (−2) = 9 and y = (1 × 4 − 1 × 14) ÷ (−2) = 5
- 2x² − 5x + 3 = 0: D = 25 − 24 = 1, so there are two distinct real roots
- x = (5 ± 1) ÷ 4, so x = 3/2 or x = 1
Nature of quadratic roots
How the discriminant decides the roots, with examples.
| Equation | D = b² − 4ac | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | 1 | 3, 2 |
| 2x² − 5x + 3 = 0 | 1 | 3/2, 1 |
| x² − 4x + 4 = 0 | 0 | 2 |
| x² + 2x + 5 = 0 | -16 | D < 0: no real roots |
D < 0 roots exist only as complex numbers.