A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student using the gym observed that when she used the badminton court for 10 hours, her bill was ₹800. When she used it for 15 hours, her bill was ₹1100. If the monthly bill y depends on the hours of the use of the badminton court, x, according to the relation y = ax + b, find the values of a and b.
a = 60, b = 200
Explanation
The problem requires finding the coefficients 'a' and 'b' in the linear equation y = ax + b. By substituting the given pairs of values for x (hours) and y (bill), we create a system of two linear equations. Following the substitution method illustrated in the textbook context for similar linear relationship problems, we isolate 'b' from the first equation and substitute it into the second to find 'a'. Once 'a' is found, we substitute it back to find 'b'. In this context, 'a' represents the variable cost per hour and 'b' represents the fixed monthly fee.
Solution Steps
Step 1: Substitute the given values into the relation y = ax + b. When x = 10 and y = 800, we get 800 = 10a + b. When x = 15 and y = 1100, we get 1100 = 15a + b.
Step 2: Express b in terms of a from the first equation. b = 800 - 10a.
Step 3: Substitute the value of b into the second equation. 1100 = 15a + (800 - 10a).
Step 4: Simplify the equation to find the value of a. 1100 = 5a + 800, which gives 5a = 300. Therefore, a = 60.
Step 5: Substitute the value of a back into the expression for b. b = 800 - 10(60) = 800 - 600. Therefore, b = 200.