# How do you the equation of the line that passes through point (4,2) and (6,6)?

Oct 4, 2016

$\textcolor{m a \ge n t a}{2 x - y = 6}$

#### Explanation:

The slope of the line through $\left(4 , 2\right)$ and $\left(6 , 6\right)$
is $\textcolor{g r e e n}{m} = \frac{\Delta y}{\Delta x} = \frac{6 - 2}{6 - 4} = \frac{4}{2} = \textcolor{g r e e n}{2}$

The slope-point form for a line
with slope $\textcolor{g r e e n}{m}$
through the point $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$
is $y - \textcolor{b l u e}{b} = \textcolor{g r e e n}{m} \left(x - \textcolor{red}{a}\right)$

Using $\left(\textcolor{red}{4} , \textcolor{b l u e}{2}\right)$ as our point (we could have used either of the given points)
and the previously determined $\textcolor{g r e e n}{m = 2}$

The slope-point form for the required line is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{b l u e}{2} = \textcolor{g r e e n}{2} \left(x - \textcolor{red}{4}\right)$

This could easily be converted into slope-point form as
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} x - 6$

or into standard form as
$\textcolor{w h i t e}{\text{XXX}} x - 2 y = 6$

Oct 4, 2016

$y = 2 x - 6$

#### Explanation:

The equation of a line in $\textcolor{b l u e}{\text{slope-intercept form}}$ is

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{y = m x + b} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where m represents the slope and b, the y-intercept.

To calcuate m, use the $\textcolor{b l u e}{\text{gradient formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are (4 ,2) and (6 ,6)

let $\left({x}_{1} , {y}_{1}\right) = \left(4 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(6 , 6\right)$

$\Rightarrow m = \frac{6 - 2}{6 - 4} = \frac{4}{2} = 2$

Thus partial equation is : $y = 2 x + b$

To find b, substitute either of the 2 given points into the partial equation and solve for b.

Using (4 ,2) : $\left(2 \times 4\right) + b = 2 \Rightarrow b = 2 - 8 = - 6$

$\Rightarrow y = 2 x - 6 \text{ is the equation of the line}$