Class 9 Mathematics Fundamentals
In mathematics, 'linear' refers to relationships that form a straight line when graphed. It indicates a constant rate of change.
A linear equation must have variables raised ONLY to the first power (degree 1). No exponents higher than 1 are allowed.
Unlike quadratic or cubic equations, linear equations never produce curves, parabolas, or other non-straight graphs.
Linear: 2x + 3 = 7 (straight line) Non-Linear: x² + 2x - 1 = 0 (parabola curve)
A variable is a symbol (usually a letter) that represents an unknown or changeable quantity in mathematical equations.
Variables act as placeholders for values we aim to determine or express relationships between changing quantities.
Typically denoted by letters like x, y, z. In two-variable equations, x and y are most frequently used.
In 5x + 2 = 12, x is the variable representing the unknown number that makes the equation true.
A linear equation is a statement of equality between two algebraic expressions with degree 1.
ax + b = 0 where a ≠ 0 (a and b are constants, x is variable)
The equation maintains balance – operations performed on one side must be mirrored on the other.
Equation: 3x - 6 = 0 Solution: x = 2 (Verified: 3×2 - 6 = 0)
An equation expressible as ax + by + c = 0 where a, b, c are real numbers (a and b not both zero), with x and y as variables.
Both variables must have exponents of exactly 1, and their product (xy term) must not appear.
Solutions are ordered pairs (x, y) that satisfy the equation. Infinite solutions exist for a single equation.
2x + 3y - 6 = 0 (a=2, b=3, c=-6)
x + y = 10 Solutions: (4,6), (5,5), (10,0) (Each pair sums to 10)
3x - 2y = 6 Solutions: (2,0), (4,3) (Verified: 3×2 - 2×0 = 6, 3×4 - 2×3 = 6)
Cost Problems: If burgers cost ₹x and fries cost ₹y, 2x + 3y = 120 represents total cost for 2 burgers and 3 fries.
Every linear equation in two variables corresponds to a unique straight line on the Cartesian plane.
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