Algebra / Systems Linear Equations
Parameterized linear systems
Parameters in linear systems usually affect slopes, intercepts, or constants. Compare coefficient ratios to decide whether lines intersect once, never, or infinitely many times.
Least you need to know
- Exactly one solution means different slopes.
- No solution means parallel distinct lines.
- Infinitely many solutions means the equations represent the same line.
Key notation
- k — A parameter whose value can change the system
Worked example
For $2x+y=5$ and $4x+2y=c$, the second left side is twice the first. The system has infinitely many solutions when $c=10$ and no solution when $c\ne 10$.
Common mistakes
- Compare constants as well as variable coefficients.
- A parameter value can change the number of solutions.
How to recognize it
- The question asks for no solution, one solution, or infinitely many solutions.
- Two equations have proportional $x$- and $y$-coefficients.
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