Unique solution and intersecting lines
A pair of linear equations has a unique solution when the two lines intersect at exactly one point.
In one line: One crossing point means one answer pair.
Practice This ConceptUnderstand it
When two lines cross at one point, that point is the only ordered pair that satisfies both equations. This happens when the equations are consistent and independent. In algebraic form, the coefficients do not make the equations multiples of each other.
Example
Formula and rule
For one unique solution, the two lines must not be parallel and must not coincide.
In ratio form, a1/a2 is not equal to b1/b2.
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
For x + y = 5 and x - y = 1, adding gives 2x = 6, so x = 3.
- 1Substituting gives y = 2.
- 2The lines meet at (3, 2), so the solution is unique.
Method you can reuse
- 1Compare the equations or draw the graph.
- 2Check whether the lines intersect once.
- 3Solve algebraically if required.
- 4Write the ordered pair.
- 5Verify both equations.
Sketch it in your answer
Show two straight lines crossing once, and clearly mark the intersection point with coordinates.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
What kind of graph relation gives exactly one solution for a pair of linear equations?
Think first, then show answerHide answer
Common wrong answer
A common wrong answer is to call every intersecting pair 'infinite'.
Why it loses marks: Only one meeting point means only one solution.
Don't mix this up
Exam tip
Write it in the exam
Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.
How CBSE asks it
- 1
Enough for 1 mark
A pair of linear equations has a unique solution when the two lines intersect at exactly one point.
- 2
Add for 2 marks
Example: The equations x + y = 5 and x - y = 1 intersect at one point, so they have a unique solution.
- 3
Add for 3 marks
When two lines cross at one point, that point is the only ordered pair that satisfies both equations.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
Help improve this page
Found something confusing, incorrect, or missing?