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P=(2)/(5)m+1.5
The price,
P, in dollars, of a train ticket used to travel a distance of
m miles is given by the equation. By how many miles does the traveling distance increase if the ticket price increases by 1 dollar?

$P=\frac{2}{5} m+1.5$ $\newline$ The price, $P$ , in dollars, of a train ticket used to travel a distance of $m$ miles is given by the equation. By how many miles does the traveling distance increase if the ticket price increases by $1$ dollar?

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Write linear functions: word problems

Full solution

Q.

P=\frac{2}{5} m+1.5

\newline

The price,

P

, in dollars, of a train ticket used to travel a distance of

m

miles is given by the equation. By how many miles does the traveling distance increase if the ticket price increases by

1

dollar?

Understand equation and question: Understand the given equation and what is asked. $\newline$ The equation $P = \frac{2}{5}m + 1.5$ represents the price of a train ticket in dollars for traveling a distance of $m$ miles. We need to find out how much the distance $m$ increases when the price $P$ increases by $1$ dollar.
Set up equation for price increase: Set up the equation to represent the increase in price. $\newline$ Let's say the initial price is $P$ dollars for $m$ miles. If the price increases by $1$ dollar, the new price will be $P + 1$ dollars for $m + \Delta m$ miles, where $\Delta m$ is the increase in distance we want to find.
Write equation for new price: Write the equation for the new price. $\newline$ Using the given equation, the new price $P + 1$ can be represented as: $\newline$ $P + 1 = \left(\frac{2}{5}\right)(m + \Delta m) + 1.5$
Substitute original price: Substitute the original price back into the equation. $\newline$ We know that the original price $P$ is $\frac{2}{5}m + 1.5$ , so we can substitute this back into the equation from Step $3$ to get: $\newline$ $\frac{2}{5}m + 1.5 + 1 = \frac{2}{5}(m + \Delta m) + 1.5$
Simplify the equation: Simplify the equation. $\newline$ Now we simplify the equation by subtracting $(\frac{2}{5})m + 1.5$ from both sides to isolate $\Delta m$ : $\newline$ $1 = (\frac{2}{5})\Delta m$
Solve for $\Delta m$ : Solve for $\Delta m$ .
To find $\Delta m$ , we divide both sides of the equation by $(2/5)$ :
$\Delta m = \frac{1}{(2/5)}$
$\Delta m = 1 \times \left(\frac{5}{2}\right)$
$\Delta m = \frac{5}{2}$
$\Delta m = 2.5$

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