Q.A shopkeeper's monthly electricity bill $B$ (in rupees) is a linear function of the number of units $u$ consumed. When $200$ units are used the bill is ₹1500, and when $350$ units are used the bill is ₹2250. Express $B$ as a function of $u$, and hence find the fixed monthly charge and the bill for $400$ units.

A linear function through two known points is determined exactly like the equation of a straight line: find the rate per unit, then the fixed part.

Since $B$ is linear in $u$, write $B(u)=mu+c$, where $m$ is the charge per unit and $c$ is the fixed monthly charge. We are given two points on the line: $(200,1500)$ and $(350,2250)$.

Slope (charge per unit):

So each unit costs ₹5, and $B(u)=5u+c$.

Fixed charge. Substitute $(200,1500)$:

Hence

The fixed monthly charge is $c=B(0)=$ ₹500.

Bill for $400$ units:

$B(400)=5(400)+500=2000+500=2500.$ …

A common slip is computing the slope with the coordinates subtracted in inconsistent order (e.g. $\frac{2250-1500}{200-350}$), which flips the sign of $m$. Keep the bill-differences and unit-differ …