The point-slope equation of the line through (6,4), and (7,2) is given as:[tex]y-2 = -2(x -7)[/tex]
Given that:
- The line passes through: (6,4) and (7,2)
To find:
The point-slope equation of that straight line.
The point slope form:
The point slope form of a straight line typically looks like:
[tex]y - y_1 = m(x - x_1)[/tex]
Formation of the equation:
Slope ( represented by m, say) can be calculated by the evaluating the ratio of how much growth happened in how much difference in the x axis.
Here, the point are (6,4) and (7,2)
They represent:
as x goes from 6 to 7 (which means 1 unit growth),
the y goes from 4 to 2 (which means opposite 2 units ie -2 unit growth)
Thus, slope will be their ratio as:
[tex]m = \dfrac{-2}{1} = -2[/tex]
The line equation having slope m is represented as y = mx + c.
Putting coordinate values (6,4) to evaluate the value of intercept c:
[tex]y = mx + c\\4 = -2 \times 6 + c\\4 = -12 + c\\c = 12 + 4 = 16[/tex]
Thus, the equation of the considered line is: y = -2x + 16.
Converting it in the point slope intercept form, with [tex]y_1 = 2[/tex]
[tex]y = -2x + 16\\y - 2 = -2x + 16 - 2\\y-2 = -2x + 14\\y - 2 = -2(x - 7)[/tex]
Thus, the point-slope equation of the given line is: [tex]y-2 = -2(x -7)[/tex]
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