Step 1: Set up the two equations by cross-checking each time frame.
Let Adil's present age be $x$ years and let Bharat's present age be $y$ years.
Five years ago, Adil's age was $x - 5$ and Bharat's age was $y - 5$. The question says Adil was thrice as old as Bharat at that time, so:
\[ x - 5 = 3(y - 5) \]
\[ x - 5 = 3y - 15 \]
\[ x - 3y + 10 = 0 \]
This is Equation 1.
Ten years later, Adil's age will be $x + 10$ and Bharat's age will be $y + 10$. The question says Adil will be twice as old as Bharat then, so:
\[ x + 10 = 2(y + 10) \]
\[ x + 10 = 2y + 20 \]
\[ x - 2y - 10 = 0 \]
This is Equation 2.
Step 2: Check consistency using the determinant of the coefficients.
Write down the coefficients of $x$ and $y$ from both equations: for Equation 1, $a_1 = 1$, $b_1 = -3$; for Equation 2, $a_2 = 1$, $b_2 = -2$.
Two lines meet at exactly one point (a unique solution exists) when the value $a_1 b_2 - a_2 b_1$ is not zero.
\[ a_1 b_2 - a_2 b_1 = (1)(-2) - (1)(-3) = -2 + 3 = 1 \]
Since this value is $1$, which is not zero, the two lines are not parallel and not the same line, so they cross at one point. The system is consistent and has a unique solution.
Step 3: Solve the pair of equations using cross multiplication.
For two equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, the cross multiplication rule gives:
\[ \frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1} \]
Here $a_1 = 1, b_1 = -3, c_1 = 10$ and $a_2 = 1, b_2 = -2, c_2 = -10$.
Work out each part one at a time.
\[ b_1c_2 - b_2c_1 = (-3)(-10) - (-2)(10) = 30 + 20 = 50 \]
\[ c_1a_2 - c_2a_1 = (10)(1) - (-10)(1) = 10 + 10 = 20 \]
\[ a_1b_2 - a_2b_1 = (1)(-2) - (1)(-3) = -2 + 3 = 1 \]
So:
\[ \frac{x}{50} = \frac{y}{20} = \frac{1}{1} \]
This gives $x = 50$ and $y = 20$ directly, without eliminating one variable by subtracting the equations.
Final Answer:
(i) The two equations are $x - 3y + 10 = 0$ and $x - 2y - 10 = 0$.
(ii) Since $a_1b_2 - a_2b_1 = 1 \neq 0$, the system is consistent and has a unique solution.
(iii) Adil's present age is 50 years and Bharat's present age is 20 years.
\[ \boxed{x = 50, \ y = 20} \]