Class 10 Linear Equations: One, No or Infinite Solutions

Read the pair as two lines

Swipe across the table to see all columns.

See the shared points

Each graph uses the same axes and unit scale. An intersection is a pair satisfying both equations, two distinct parallel lines share none, and coincident lines share every point on the line. The dashed brown overlay in the last graph distinguishes the second equation without moving its line.

Understand what elimination tells you

For the intersecting pair, adding equations gives 2x = 8, so x = 4 and y = 3. Substitution into both original equations confirms the shared point.

For the parallel pair, doubling the first gives 2x + 2y = 14, while the second demands 18. Subtraction produces 0 = 4, an impossibility. This is a statement about consistency, not a request to divide by zero.

For the coincident pair, subtraction gives 0 = 0. It does not force x and y to be zero. Every pair on x + y = 7 works, including (0,7), (3,4) and (7,0).

Use ratios with care

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the familiar coefficient-ratio tests work when the required denominators are nonzero. If a denominator is zero, inspect proportionality directly or eliminate a variable rather than writing an undefined ratio.

A vertical line such as x = 4 is still a valid line. Rewriting every equation as y = mx + c is unnecessary and fails for this case. Two distinct vertical lines are parallel; a vertical and a nonvertical line intersect once.

Use this page before practice

Sort a pair into one of the three outcomes, then solve only when appropriate. On graph paper, label axes, use a consistent scale and check the intersection algebraically if an exact value is required. Approximate plotting should not silently replace an exact fraction.

Choose what to study next

Sources and scope

This is a focused learning resource, not a complete chapter or a prediction of board questions. Use your current syllabus and the paper-specific marking scheme for assessment requirements.

Try a short check, then return to the explanation

Optional TutorMax original practice: five questions on linear equations, with explanations and a fresh retest. These checks sample a few skills; they are not official paper questions or a whole-subject readiness score. Existing explanations and official paper downloads stay available on this page.