A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was ₹400. When she accessed 14 modules, her bill was ₹500. If the monthly bill y depends on the number of modules accessed, x, according to the relation y = ax + b, find the values of a and b.
a = 25, b = 150
Explanation
Following the substitution method illustrated in the textbook's Example 11, we formulate two linear equations based on the student's observations. By expressing the fixed fee (b) in terms of the variable cost (a) from the first condition and substituting it into the second condition, we can solve for 'a'. Once 'a' is found, it is substituted back to find the value of 'b'.
Solution Steps
Step 1: Substitute the given values into the relation y = ax + b. For 10 modules (x = 10, y = 400), we get 400 = 10a + b. For 14 modules (x = 14, y = 500), we get 500 = 14a + b.
Step 2: Express b in terms of a from the first equation: b = 400 - 10a.
Step 3: Substitute this value of b into the second equation: 500 = 14a + (400 - 10a).
Step 4: Solve for a: 500 = 4a + 400, which simplifies to 100 = 4a, so a = 25.
Step 5: Substitute the value of a back into the expression for b: b = 400 - 10(25) = 400 - 250 = 150.