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Let \(A\) and \(B\) be two distinct points whose coordinates are \((x_1,y_1)\) and \((x_2,y_2)\), respectively. The slope of the straight line passing through the points \(A\) and \(B\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where (\(x_1 \neq x_2\)).
Substitute the value of \(m\) in point slope form, we get:
\(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\)
\(\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\)
Therefore, the equation of the line in two point form is \(\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\).
Example:
Find the equation of the straight line passing through the points \((-1,6)\) and \((2,5)\).
Solution:
Here, \((x_1,y_1) = (-1,6)\)
\((x_2,y_2) = (2,5)\)
Substituting the known values in the formula \(\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\), we get:
\(\frac{y - 6}{5 - 6} = \frac{x + 1}{2 - 6}\)
\(\frac{y - 6}{-1} = \frac{x + 1}{-4}\)
\(-4(y - 6) = -1(x + 1)\)
\(-4y + 24 = -x - 1\)
\(x - 4y + 25 = 0\)
Therefore, the equation of the straight line passing through the points \((-1,6)\) and \((2,5)\) is \(x - 4y + 25 = 0\).
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