Question

How do you find a general form equation for the line through the pair of points $(1,2)$ and $(5,4)?$

Answer 1

To solve such a type of question we will use the standard formula for finding line equations when two pints on the line are given. Here, We have two points given and we know two point form of line so we will write an equation of line using these two given points.

Complete step by step solution:
we have $(1,2)$,$(5,4)$ two points of a line if two points are $({x_1},{y_1})$,$({x_2},{y_2})$ is
$y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})$
Using this formula or form of line .We will write an equation of line.
Here in this question,
$({x_1} = 1,{y_1} = 2),({x_2} = 5,{y_2} = 4)$
$y - 2 = \dfrac{{4 - 2}}{{5 - 1}}(x - 1)$
$\Rightarrow y - 2 = \dfrac{2}{4}(x - 1)$
$\Rightarrow 2y - 4 = x - 1$
$\Rightarrow 2y - x - 3 = 0$
So, here line is $2y - x - 3 = 0$ or $y = \dfrac{1}{2}x + \dfrac{3}{2}$
So the equation of the line which goes through the point $(1,2)$ and $(5,4)$ is; $y = \dfrac{1}{2}x + \dfrac{3}{2}$. The equation is now in $y = mx + b$ form (which is slope-intercept form) where the slope is $m = \dfrac{1}{2}$ and the y-intercept is $b = \dfrac{3}{2}$

Note: In such types of questions students may confuse while putting both given points into the equation of the line. so always try to focus when you put values into the equation and verify is it ok or not. Now you are also able to solve for the y-intercept and slope of any line through the pair of points is given.