Homeclass 10MathsPair of Linear Equations in Two VariablesConsistent System

Consistent System

A consistent system of linear equations in two variables is a pair of equations that has at least one valid solution. Graphically, this means the lines representing the equations either intersect at a single point or overlap completely.

Key Concepts

If the two lines intersect at exactly one point, the system is consistent and has a unique solution.
If the two lines are coincident or overlap entirely, the system is consistent and has infinitely many solutions.
Consistency can be determined algebraically by comparing the ratios of the coefficients of the variables and the constant terms without drawing the graph.

Formula & Equation

For equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, the system is consistent if a1/a2 != b1/b2 (unique solution) or a1/a2 = b1/b2 = c1/c2 (infinitely many solutions).

a1 and a2 are coefficients of x, b1 and b2 are coefficients of y, and c1 and c2 are the constant terms.

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Common Misconceptions

Myth: A consistent system always means there is only one unique solution.
Fact: A consistent system means there is at least one solution. It includes both intersecting lines which have exactly one solution, and coincident lines which have infinitely many solutions.

Real World Applications

Finding the exact meeting point of two vehicles traveling on different straight roads that cross each other at a specific junction.
Comparing two mobile data plans that charge the exact same monthly rental and per-gigabyte rate, meaning their cost graphs overlap perfectly for any amount of data used.

Frequently Asked Questions

How do you check if a pair of linear equations is consistent?
Compare the ratios of their coefficients. If a1/a2 is not equal to b1/b2, or if a1/a2 = b1/b2 = c1/c2, the system is consistent.
Can a consistent system have parallel lines?
No, parallel lines never intersect, meaning they have no solution. A system with parallel lines is called an inconsistent system.

Keep Learning

Intersecting LinesCoincident LinesDependent System