Introduction to Linear Polynomials | Exercise 2.5

Question 2

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 $b$ .

Solution

Understand the Question

The monthly bill $y$ and hours of court use $x$ follow the linear relation $y = ax + b$ , where $a$ is the hourly rate and $b$ is the fixed monthly fee.
We are given two pairs of values: $(x_1, y_1) = (10, 800)$ and $(x_2, y_2) = (15, 1100)$ .
Substituting these pairs into the relation gives a system of two linear equations in $a$ and $b$ , which can be solved using elimination.

Step 1 · Formulate Linear Equations

Given the relation $y = ax + b$

For $x = 10$ and $y = 800$

\begin{aligned} 800 &= a(10) + b \\[0.6em] 10a + b &= 800 \quad \dots (1) \end{aligned}

For $x = 15$ and $y = 1100$

\begin{aligned} 1100 &= a(15) + b \\[0.6em] 15a + b &= 1100 \quad \dots (2) \end{aligned}

Step 2 · Solve for $a$

Subtract equation $(1)$ from equation $(2)$

\begin{aligned} (15a + b) - (10a + b) &= 1100 - 800 \\[0.6em] 15a - 10a + b - b &= 300 \\[0.6em] 5a &= 300 \\[0.6em] a &= \dfrac{300}{5} \\[0.6em] a &= 60 \end{aligned}

Step 3 · Solve for $b$

Substitute $a = 60$ into equation $(1)$

\begin{aligned} 10(60) + b &= 800 \\[0.6em] 600 + b &= 800 \\[0.6em] b &= 800 - 600 \\[0.6em] b &= 200 \end{aligned}

Answer

$a = 60$ and $b = 200$

Common Mistakes

Swapping Variables: Confusing the independent variable $x$ (hours) with the dependent variable $y$ (total bill).
Sign Errors in Subtraction: Forgetting to distribute the negative sign to both terms when calculating $(15a + b) - (10a + b)$ .

More questions in Exercise 2.5

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 $b$ .

Consider the relationship between temperature measured in degrees Celsius ( $^\circ\text{C}$ ) and degrees Fahrenheit ( $^\circ\text{F}$ ), which is given by $^\circ\text{C} = a^\circ\text{F} + b$ . Find $a$ and $b$ , given that ice melts at $0$ degrees Celsius and $32$ degrees Fahrenheit, and water boils at $100$ degrees Celsius and $212$ degrees Fahrenheit. (Hint: When $^\circ\text{C} = 0$ , $^\circ\text{F} = 32$ and when $^\circ\text{C} = 100$ , $^\circ\text{F} = 212$ . Use this information to find $a$ and $b$ , and thus, the linear relationship between $^\circ\text{C}$ and $^\circ\text{F}$ .)