(a) The sum of the digits of a 2-digit number is 12. Seven times the number is equal to four times the number obtained by reversing the order of the digits. Find the number. OR (b) Find the values of x and y from the following pair of linear equations : $\frac{x}{2} + y = 0.8$ $\frac{7}{x + y} = 10$

Step 1: Set up equations for the 2-digit number problem

Let the two-digit number be represented as $10t + u$, where $t$ is the tens digit and $u$ is the units digit. The first condition states that the sum of the digits is 12, which gives us the equation $t + u = 12$. The second condition states that seven times the number is equal to four times the number obtained by reversing the digits. Reversing the digits gives the number $10u + t$. This leads to the equation $7(10t + u) = 4(10u + t)$.

Step 2: Simplify the second equation

Expand both sides of the second equation: $70t + 7u = 40u + 4t$. Rearrange the terms to group $t$ and $u$ on opposite sides: $70t - 4t = 40u - 7u$, which simplifies to $66t = 33u$. Dividing both sides by 33 gives us a simpler relationship between $t$ and $u$: $2t = u$.

Step 3: Solve the system of equations

Now we have a system of two linear equations: $t + u = 12$ and $u = 2t$. Substitute the expression for $u$ from the second equation into the first equation: $t + (2t) = 12$. This simplifies to $3t = 12$, so $t = 4$. Substitute the value of $t$ back into $u = 2t$ to find $u = 2(4) = 8$.

Step 4: Form the number

With $t = 4$ and $u = 8$, the original two-digit number is $10t + u = 10(4) + 8 = 40 + 8 = 48$.

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