Problem:
A fraction becomes 1/4 when 2 is subtracted from the numerator and 3 is added to the denominator. Represent this as a linear equation in two variables.

Solution:

To represent the given problem as a linear equation in two variables, we need to assign variables to the numerator and denominator of the fraction.

Let's assume the numerator of the fraction is 'x' and the denominator is 'y'.

So, the original fraction can be represented as x/y.

According to the problem, when 2 is subtracted from the numerator and 3 is added to the denominator, the new fraction becomes 1/4.

This can be expressed as (x - 2) / (y + 3) = 1/4.

Now, let's simplify this equation further.

Simplifying the equation:
To solve the equation, we can cross-multiply the fractions.

4(x - 2) = 1(y + 3)

Now, let's distribute and simplify.

4x - 8 = y + 3

To represent this equation in the standard form (Ax + By = C), we need to rearrange it.

Subtracting 'y' from both sides:

4x - y - 8 = 3

Rearranging the equation:

4x - y = 3 + 8

4x - y = 11

Therefore, the linear equation in two variables representing the given problem is 4x - y = 11.

Explanation:
In this problem, we are given a fraction that becomes 1/4 when certain operations are performed on its numerator and denominator. We assign variables 'x' and 'y' to represent the numerator and denominator respectively.

By substituting the given operations into the variables, we can form the equation (x - 2) / (y + 3) = 1/4. Simplifying this equation by cross-multiplication, we get 4x - y = 11.

This linear equation in two variables represents the relationship between the numerator and denominator of the fraction and allows us to solve for their values.