Linear Equation Problem Set 1, Q No 6, SSC 10th New Syllabus
Question 6: Solve the following simultaneous equations:
$$ \frac{2}{x} - \frac{3}{y} = 15 $$
$$ \frac{8}{x} + \frac{5}{y} = 77 $$
- A) $x = 1, y = 1$
- B) $x = \frac{1}{9}, y = 1$ ✓ Correct
- C) $x = 9, y = 1$
- D) $x = \frac{1}{3}, y = \frac{1}{5}$
Solution: Let $\frac{1}{x} = m$ and $\frac{1}{y} = n$. The given equations become linear equations in variables $m$ and $n$:
$2m - 3n = 15$ ... (I)
$8m + 5n = 77$ ... (II)
Multiply equation (I) by 4 to equate the coefficients of $m$:
$8m - 12n = 60$ ... (III)
Subtract equation (III) from equation (II):
$17n = 17 \implies n = 1$
Substitute $n = 1$ in equation (I):
$2m - 3(1) = 15 \implies 2m = 18 \implies m = 9$
Now, resubstitute the values of $m$ and $n$:
$m = \frac{1}{x} \implies 9 = \frac{1}{x} \implies x = \frac{1}{9}$
$n = \frac{1}{y} \implies 1 = \frac{1}{y} \implies y = 1$
Therefore, the solution is $(x, y) = \left(\frac{1}{9}, 1\right)$.
