Not every system of simultaneous linear equations has a unique solution. Some have no solution (parallel lines), while others have infinite solutions (coincident lines). In DSE Paper 2 MC and Paper 1 Section A(1), you may be asked to identify the type of solution or find parameter values that lead to a particular case. This article teaches you the coefficient ratio method to quickly determine the solution type, with clear examples, common traps, and plenty of practice.
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"No Solution" vs "Infinite Solutions" in Simultaneous Equations
Exercise
Determine the number of solutions or find the parameter as required.
- \(\begin{cases} x + y = 5 \\ 2x + 2y = 10 \end{cases}\)
- \(\begin{cases} 3x - 2y = 4 \\ 6x - 4y = 7 \end{cases}\)
- \(\begin{cases} 2x + 3y = 6 \\ 4x + 6y = 12 \end{cases}\)
- Find \(k\) such that \(\begin{cases} 2x + ky = 8 \\ 4x + 6y = 16 \end{cases}\) has infinite solutions.
- Find \(k\) such that \(\begin{cases} 3x + 2y = 5 \\ 6x + ky = 10 \end{cases}\) has no solution.
- Find \(k\) such that \(\begin{cases} kx - 3y = 2 \\ 4x - 6y = 4 \end{cases}\) has a unique solution.
- \(\begin{cases} 5x - 2y = 3 \\ 10x - 4y = 6 \end{cases}\)
- \(\begin{cases} 4x + 5y = 8 \\ 8x + 10y = 17 \end{cases}\)
- Find \(k\) such that \(\begin{cases} 2x - y = 1 \\ 6x - 3y = k \end{cases}\) has infinite solutions.
- Find \(k\) such that \(\begin{cases} kx + 2y = 3 \\ 3x + 4y = 6 \end{cases}\) has no solution.
Answer Key
1. Infinite
2. No solution
3. Infinite
4. \(k=3\)
5. \(k=4\)
6. \(k \neq 2\)
7. Infinite
8. No solution
9. \(k=3\)
10. \(k = \frac{3}{2}\)