Short answer: for two linear equations, compare the ratios of their coefficients. Same ratio for everything means infinitely many solutions. Same ratio for x and y but a different constant means no solution. Anything else means exactly one.
The rule
Write both equations as ax + by = c. For the system
- a₁x + b₁y = c₁
- a₂x + b₂y = c₂
the lines relate like this:
- Infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂. The equations describe the same line.
- No solution: a₁/a₂ = b₁/b₂ but c₁/c₂ is different. The lines are parallel.
- One solution: a₁/a₂ and b₁/b₂ are different. The lines cross once.
Example: infinitely many solutions
Question: In the system 3x + 2y = 7 and kx + 6y = 21, for what value of k does the system have infinitely many solutions?
The y-coefficients go from 2 to 6, which is multiplying by 3. The constants go from 7 to 21, also multiplying by 3. So the second equation must be the first times 3:
3(3x + 2y) = 3(7), which gives 9x + 6y = 21.
So k = 9.
Example: no solution
Change the second equation to kx + 6y = 14. With k = 9, the left side is still 3 times the first equation, but the right side is 14 instead of 21. The lines are parallel, so the system has no solution.
Why it is fast
You skip substitution and elimination. You compare numbers, which takes seconds. Always check the constant: matching x and y ratios alone do not tell you if the lines are the same or parallel.
Use Desmos as a check
Graph both equations. If the lines overlap, it is infinitely many. If they never meet, none. If they cross, one. Seeing it confirms the rule.
Keep going
Systems are part of the Algebra module in the course. Book a free call to see where your own points are going.