Question

A 3 – mile cab ride costs Rs.3. A 6 – mile cab costs Rs.4.80. How do you find a linear equation that models cost (c) as a function of distance (d)?

Answer 1

Hint: Assume the required linear equation as \[c=md+k\], where c and d represent the cost and distance respectively. Here ‘m’ represents the slope of the straight line and k represents the intercept on the coast axis if we draw a graph with c on the y-axis and d on x-axis. Now, form two linear equations using the provided values of c and d and find the values of m and k using them. Finally, substitute the obtained value of m and k in the assumed linear equation to get the answer.

Complete step by step answer:
Here, we have been provided with the information regarding the cost of the cab ride according to the given distance. We have been asked to form a linear equation that relates to cost (c) as a function of distance (d).
Now, whenever we assume a linear equation considering y as a function of x, we assume it as \[y=ax+b\] where ‘a’ represents the slope and ‘b’ represents the intercept on the y-axis. Here, we have to determine ‘c’ as a function of ‘d’, so ‘y’ will be replaced with ‘c’ and ‘d’ will replace ‘x’. Therefore, the required linear equation can be given as: -
\[\Rightarrow c=md+k\] - (1)
Here, we have assumed ‘m’ as the slope and ‘k’ as the intercept in place of ‘a’ and ‘b’ respectively. Now, let us consider the two conditions given, one – by – one.
1. Cost of a 3 – mile cab is Rs.3.
Here, d = 3, c = 3. So, substituting these values in equation (1), we get,
\[\Rightarrow 3=3m+k\] - (2)
2. Cost of a 6 – mile cab is Rs.4.80.
Here, d = 6, c = 4.80. So, substituting these values in equation (1), we get,
\[\Rightarrow 4.80=6m+k\] - (3)
Solving equations (3) and (2), we get,
\[\Rightarrow \] k = 1.2 and c = 0.6
Therefore, the required linear equation can be given as: -
\[\Rightarrow c=0.6d+1.2\]
Hence, the above relation is our answer.

Note:
 One may note that we have assumed the linear equation in slope-intercept form. You can assume it in standard form also given as \[\dfrac{x}{a}+\dfrac{y}{b}=1\]. Note that you must read the question carefully that we are required to assume ‘c’ as a function of ‘d’ and not ‘d’ as a function of ‘c’. So, do not assume \[d=mc+k\] as the required linear equation. You must know how to solve linear equations. At last, substitute the values of c and d carefully and according to the information provided in the question otherwise, you will get the wrong values of ‘m’ and ‘k’.