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Pair Of Linear Equations

A pair of linear equations in two variables consists of two linear equations that share the exact same two variables. Together, they form a system of simultaneous equations that geometrically represent two straight lines on a two-dimensional Cartesian plane.

Key Concepts

The standard form of a pair of linear equations in two variables x and y is a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.

Geometrically, the two lines represented by these equations can either intersect at a single point, remain strictly parallel, or completely coincide with each other.

Algebraically, the system can have a unique solution, no solution, or infinitely many solutions depending on the ratios of their respective coefficients.

Formula & Equation

a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

x and y are the variables. a1, b1, a2, and b2 are non-zero real number coefficients. c1 and c2 are the constant terms.

Common Misconceptions

Myth: Students often assume that every pair of linear equations must have exactly one solution.

Fact: A pair of linear equations can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). It depends entirely on the ratio of their coefficients.

Real World Applications

Calculating the individual fixed base fare of a taxi and the variable charge per kilometer when given the total cost of two different trips.

Determining the speed of a boat in still water and the speed of the river current using the time taken to travel upstream and downstream.

Frequently Asked Questions

What is the condition for a pair of linear equations to have a unique solution?

A pair of linear equations has a unique solution if the ratio of the coefficients of x is not equal to the ratio of the coefficients of y (a1/a2 is not equal to b1/b2).

What is the difference between consistent and inconsistent linear equations?

A consistent pair of linear equations has at least one solution (intersecting or coincident lines), whereas an inconsistent pair has no solution and represents parallel lines on a graph.

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