Find whether the following pairs of linear equations intersect, are parallel or coincide (without drawing graphs)
Video Explanation
Watch the video explanation below:
Method Used
For two linear equations:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
- If a₁/a₂ ≠ b₁/b₂ → Lines intersect
- If a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → Lines are parallel
- If a₁/a₂ = b₁/b₂ = c₁/c₂ → Lines coincide
(i) 5x − 4y + 8 = 0 and 7x + 6y − 9 = 0
Here, a₁ = 5, b₁ = −4, c₁ = 8
a₂ = 7, b₂ = 6, c₂ = −9
a₁/a₂ = 5/7
b₁/b₂ = −4/6 = −2/3
Since a₁/a₂ ≠ b₁/b₂,
The lines intersect at a point.
(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
Here, a₁ = 9, b₁ = 3, c₁ = 12
a₂ = 18, b₂ = 6, c₂ = 24
a₁/a₂ = 9/18 = 1/2
b₁/b₂ = 3/6 = 1/2
c₁/c₂ = 12/24 = 1/2
Since a₁/a₂ = b₁/b₂ = c₁/c₂,
The lines coincide.
(iii) 6x − 3y + 10 = 0 and 2x − y + 9 = 0
Here, a₁ = 6, b₁ = −3, c₁ = 10
a₂ = 2, b₂ = −1, c₂ = 9
a₁/a₂ = 6/2 = 3
b₁/b₂ = −3/−1 = 3
c₁/c₂ = 10/9
Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂,
The lines are parallel.
Final Answer
- (i) Lines intersect at a point
- (ii) Lines coincide
- (iii) Lines are parallel
Conclusion
Thus, by comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, we can determine the nature of the pair of linear equations without drawing their graphs.