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Class 10 · Chapter 3 · Mathematics

Pair of Linear Equations in Two Variables Cheat Sheet

Formulae

General form of a pair of linear equations

a1x+b1y+c1=0,a2x+b2y+c2=0a_{1}x + b_{1}y + c_{1} = 0, \qquad a_{2}x + b_{2}y + c_{2} = 0

a₁, b₁, c₁, a₂, b₂, c₂ are real numbers with a₁, b₁ not both zero and a₂, b₂ not both zero.

Unique solution (intersecting lines)

a1a2≠b1b2\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}

The lines intersect at one point and the pair is consistent.

No solution (parallel lines)

a1a2=b1b2≠c1c2\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

The lines are parallel and never meet; the pair is inconsistent.

Infinitely many solutions (coincident lines)

a1a2=b1b2=c1c2\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

The lines coincide; the pair is consistent and dependent.

Substitution method

y=c2−a2xb2⇒a1x+b1y+c1=0y = \frac{c_{2}-a_{2}x}{b_{2}} \Rightarrow a_{1}x + b_{1}y + c_{1} = 0

Express one variable in terms of the other from one equation and substitute it into the other.

Elimination method

a1b2x−a2b1x=b1c2−b2c1a_{1}b_{2}x - a_{2}b_{1}x = b_{1}c_{2} - b_{2}c_{1}

Make the coefficients of one variable equal and subtract the equations to eliminate it.

Definitions

Linear equation in two variables
An equation of the form ax + by + c = 0 where a and b are not both zero.
Solution of a pair
A pair of values (x, y) that satisfies both equations at the same time.
Consistent pair
A pair of equations that has at least one solution (unique or infinitely many).
Inconsistent pair
A pair of equations that has no solution.
Dependent pair
A consistent pair with infinitely many solutions whose graphs coincide.
Graphical method
Solving a pair by drawing the two lines and reading off their point of intersection.
Substitution method
A method in which one variable is replaced by an expression in the other variable.
Elimination method
A method in which one variable is removed by adding or subtracting suitable multiples of the equations.
Coefficient
The number multiplying a variable in an equation.

Key Points

  • A pair of linear equations can be solved by the graphical method, the substitution method or the elimination method.
  • Geometrically, two lines can intersect at one point (unique solution), be parallel (no solution) or coincide (infinitely many solutions).
  • The ratios a₁/a₂, b₁/b₂ and c₁/c₂ decide the number of solutions.
  • If a₁/a₂ ≠ b₁/b₂, the equations are consistent with a unique solution.
  • If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and the pair is inconsistent.
  • If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines coincide and there are infinitely many solutions.
  • The graph of every linear equation in two variables is a straight line.
  • A pair of equations can also be applied to word problems by naming the unknowns and forming two equations.