Linear Equations

Overview

Linear equations form the backbone of algebra and appear consistently in MP TET Varg-2 mathematics sections. A linear equation is an algebraic equation in which the highest power of the variable is 1—no squares, cubes, or higher powers. These equations represent straight lines when graphed on a coordinate plane.

For the MP TET exam, you must master solving equations in one variable (finding a single unknown value) and in two variables (finding pairs of values or graphing lines). Questions typically test your ability to form equations from word problems, solve them using standard methods, and interpret solutions graphically. This topic connects directly to real-world applications like age problems, profit-loss scenarios, and distance-time relationships commonly asked in the exam.

Understanding linear equations also prepares you for teaching upper-primary students, where building conceptual clarity about variables, constants, and the balance principle of equations is essential.

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Key Concepts

• **Linear equation in one variable**: An equation of the form ax + b = 0, where a ≠ 0. It has exactly one solution (one root).

• **Linear equation in two variables**: An equation of the form ax + by + c = 0, where a and b are not both zero. It has infinitely many solutions, each represented as an ordered pair (x, y).

• **Solution of an equation**: A value (or pair of values) that makes the equation true when substituted for the variable(s).

• **Graph of linear equation in two variables**: Always a straight line. Every point on this line is a solution of the equation.

• **System of linear equations**: Two or more linear equations considered together. The solution is the point(s) where their graphs intersect.

• **Consistent system**: Has at least one solution (lines intersect or coincide). **Inconsistent system**: Has no solution (parallel lines).

• **Transposition rule**: When moving a term from one side of an equation to the other, change its sign.

• **Balance principle**: Whatever operation you perform on one side of an equation, you must perform the same on the other side.

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Standard form (one variable) | ax + b = 0, solution: x = −b/a | | Standard form (two variables) | ax + by + c = 0 | | Slope-intercept form | y = mx + c, where m = slope, c = y-intercept | | Slope from two points | m = (y₂ − y₁)/(x₂ − x₁) | | Condition for parallel lines | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (no solution) | | Condition for coincident lines | a₁/a₂ = b₁/b₂ = c₁/c₂ (infinite solutions) | | Condition for intersecting lines | a₁/a₂ ≠ b₁/b₂ (unique solution) | | Substitution method | Express one variable in terms of the other, then substitute | | Elimination method | Add or subtract equations to eliminate one variable |