Consistent and inconsistent pairs

Two straight lines can meet in three ways. They can cross at one point, which gives one solution. They can be parallel and never meet, which gives no solution. Or they can lie on top of each other, which gives endless solutions. A pair whose lines lie on top of each other is called dependent. It is still consistent.

You can tell the case without drawing. Write the equations as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Then compare the ratios of their coefficients.

A small example

• a₁/a₂ ≠ b₁/b₂: the lines cross. One solution, so consistent.
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂: the lines are parallel. No solution, so inconsistent.
• a₁/a₂ = b₁/b₂ = c₁/c₂: the lines are the same. Endless solutions, so dependent and consistent.

Take x - y - 6 = 0 and 3x - 3y - 18 = 0. The three ratios are 1/3, -1/-3 and -6/-18. All equal 1/3, so the lines are the same and the pair is dependent.

Where marks go

Question 3 of CBSE's 2026-27 Maths Standard sample paper uses this pair. One line is x - y = 6, and the pair has only one point of meeting, (5, -1). The option 3x - 3y = 18 passes through (5, -1) too, but it is the same line. The board's Maths Standard marking scheme for 2026-27 rules it out as consistent and dependent.

Section A questions carry 1 mark each, with no part marks. So a check that stops at "the point fits" earns nothing here. The ratio test is the step that finds the right option. Our analysis of the 2026 Maths Basic board paper shows how each section of a real paper was built.

About the author

Mohit Sardana

Chief Architect

An engineer (B.Tech, Computer Science) with over 20 years of preparing students for entrance exams, including as Dean at FIITJEE and Vice-President at Aakash. He is the chief architect of the TRUpreBoards evaluation engine.

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