Introduction to Linear Polynomials | Exercise 2.5

Question 1

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 $b$ .

Solution

Understand the Question

The total monthly bill $y$ and the number of modules $x$ follow the linear relation $y = ax + b$ , where $a$ represents the cost per module and $b$ is the fixed monthly fee.
Using the two given conditions $(x = 10, y = 400)$ and $(x = 14, y = 500)$ , we form a system of two linear equations in two variables ( $a$ and $b$ ).
We solve the system by elimination to find the values of $a$ and $b$ .

Step 1 · Form the Linear Equations

Given linear relation: $y = ax + b$ Diagram 1

For $x = 10$ and $y = 400$ :

\begin{aligned} 400 &= a(10) + b \\ 10a + b &= 400 \quad \dots (1) \end{aligned}

For $x = 14$ and $y = 500$ :

\begin{aligned} 500 &= a(14) + b \\ 14a + b &= 500 \quad \dots (2) \end{aligned}

Step 2 · Find the Value of $a$

Subtract Equation $(1)$ from Equation $(2)$ :

\begin{aligned} (14a + b) - (10a + b) &= 500 - 400 \\ 14a - 10a + b - b &= 100 \\ 4a &= 100 \\[0.6em] a &= \dfrac{100}{4} \\ a &= 25 \end{aligned}

Step 3 · Find the Value of $b$

Substitute $a = 25$ into Equation $(1)$ :

\begin{aligned} 10a + b &= 400 \\ 10(25) + b &= 400 \\ 250 + b &= 400 \\ b &= 400 - 250 \\ b &= 150 \end{aligned}

Answer

$a = 25$ , $b = 150$

Common Mistakes

Swapping Variables: Confusing $x$ (number of modules) and $y$ (total bill) by writing $10 = 400a + b$ instead of $400 = 10a + b$ .
Sign Error in Elimination: Forgetting to distribute the minus sign to both terms when subtracting: $(14a + b) - (10a + b) = 14a - 10a + b - b = 4a$ .

More questions in Exercise 2.5

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 $b$ .

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 $^\circ\text{F}$ .)