Examples of Two Points Form Equation of Line

Example 1:
A milkman can sell 650 liters of milk at Dollars 3.15 per liter and 800 liters of milk at Dollars 3.00 per liter. Assuming the graph of the sale price and the milk sold to be a straight line, find the number of liters of milk that the milkman can sell at Dollars 2.50 per liter.
Solution:
Let x be the number of liters of sold milk and y be the per liter price. The given information can be written in the form of points \left( {650,3.15} \right) and \left( {800,3.00} \right). Since the graph of sale price and the milk sold is a straight line, so we find the equation of straight line through the points \left(  {650,3.15} \right) = \left( {{x_1},{y_1}} \right) and \left( {800,3.00} \right) = \left( {{x_2},{y_2}}  \right) as follows:
Using the two points form of equation of a straight line

\begin{gathered} \frac{{y - {y_1}}}{{{y_2} - {y_1}}} =  \frac{{x - {x_1}}}{{{x_2} - {x_1}}} \\ \Rightarrow \frac{{y - 3.15}}{{3.00 - 3.15}}  = \frac{{x - 650}}{{800 - 650}} \\ \Rightarrow \frac{{y - 3.15}}{{ - 0.15}} =  \frac{{x - 650}}{{150}} \\ \end{gathered}


In order to fine the number of liters of milk that the milkman can sell at Dollars2.50, put y = 2.50 in the above equation, we get

\begin{gathered} \Rightarrow \frac{{2.50 - 3.15}}{{ - 0.15}}  = \frac{{x - 650}}{{150}} \\ \Rightarrow x - 650 = 4.34 \times 150 \\ \Rightarrow x = 1301 \\ \end{gathered}


This result shows that the milkman can sell 1301 liters of milk at Dollars 2.50 per liter.

Example 2:
Find the equation of straight line passing through the points A\left( {0,8} \right) and B\left( {2,3} \right).
Consider the points A\left(  {0,8} \right) = \left( {{x_1},{y_1}} \right) and B\left( {2,3} \right) = \left( {{x_2},{y_2}}  \right), now using these points in two point form of equation of straight line, we get

\begin{gathered} \frac{{y - {y_1}}}{{{y_2} - {y_1}}} =  \frac{{x - {x_1}}}{{{x_2} - {x_1}}} \\ \Rightarrow \frac{{y - 8}}{{3 - 8}} =  \frac{{x - 0}}{{2 - 0}} \\ \Rightarrow \frac{{y - 8}}{{ - 5}} =  \frac{x}{2} \\ \Rightarrow 2\left( {y - 8} \right) =  - 5x \\ \Rightarrow 5x + 2y - 16 = 0 \\ \end{gathered}


Which is the required equation of straight line.

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