Simultaneous equations often contain fractions or decimals, which can make them look intimidating. However, the key is to clear the fractions or decimals first by multiplying each equation by the LCM of denominators or by a power of 10. This transforms the system into one with integer coefficients, which you can then solve using substitution or elimination. This article provides a step-by-step strategy for handling these equations, with worked examples, common pitfalls, and plenty of DSE-style practice.
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Handling Simultaneous Equations with Fractions or Decimals
Exercise
Solve each system. Clear fractions or decimals first.
- \(\begin{cases} \frac{x}{2} + \frac{y}{3} = 6 \\ \frac{x}{3} - \frac{y}{2} = 1 \end{cases}\)
- \(\begin{cases} \frac{3x}{4} - \frac{y}{2} = 2 \\ \frac{x}{6} + \frac{y}{8} = 1 \end{cases}\)
- \(\begin{cases} 0.2x + 0.5y = 1.3 \\ 0.3x - 0.2y = 0.4 \end{cases}\)
- \(\begin{cases} 0.05x - 0.02y = 0.01 \\ 0.03x + 0.04y = 0.02 \end{cases}\)
- \(\begin{cases} \frac{2}{x} + \frac{1}{y} = 5 \\ \frac{1}{x} - \frac{3}{y} = 2 \end{cases}\)
- \(\begin{cases} \frac{3}{x} + \frac{4}{y} = 10 \\ \frac{2}{x} - \frac{5}{y} = 1 \end{cases}\)
- \(\begin{cases} 0.4x + 0.3y = 0.5 \\ 0.2x - 0.1y = 0.1 \end{cases}\)
- \(\begin{cases} \frac{x+2}{3} + \frac{y-1}{2} = 2 \\ \frac{x-1}{4} - \frac{y+3}{2} = 0 \end{cases}\)
- \(\begin{cases} \frac{5}{x} - \frac{2}{y} = 3 \\ \frac{3}{x} + \frac{4}{y} = 2 \end{cases}\)
- \(\begin{cases} 0.12x - 0.05y = 0.31 \\ 0.08x + 0.15y = 0.37 \end{cases}\)
Answer Key
1. \(x=8, y=6\)
2. \(x=4, y=2\)
3. \(x=3, y=1.4\) (or \(x=3, y=\frac{7}{5}\))
4. \(x=0.4, y=0.2\) (or \(x=\frac{2}{5}, y=\frac{1}{5}\))
5. \(x=1, y=\frac{1}{3}\)
6. \(x=\frac{1}{2}, y=\frac{1}{2}\)
7. \(x=1, y=\frac{1}{3}\)
8. \(x=5, y=-3\)
9. \(x=2, y=1\)
10. \(x=3, y=1\)