# What is the equation for the line which cuts the x-axis at 4 and goes through point (4, 2)?

Mar 5, 2017

$\left(y - \textcolor{red}{4}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{2}\right)$

Or

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{4}\right)$

Or

$y = \textcolor{red}{- 2} x + \textcolor{b l u e}{8}$

#### Explanation:

If it cuts the x-axis at $4$ this is the point $\left(4 , 0\right)$.

We can now find the slope given two points. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{0} - \textcolor{b l u e}{4}}{\textcolor{red}{4} - \textcolor{b l u e}{2}} = \frac{- 4}{2} = - 2$

We can now use the point-slope formula to write an equation for the line. The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the point from the problem gives:

$\left(y - \textcolor{red}{4}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{2}\right)$

We can also substitute the slope we calculated and the point we determine from where the 3-axis is cut giving:

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{4}\right)$

We can also solve this equation for $y$ to put the equation in slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y - \textcolor{red}{0} = \left(\textcolor{b l u e}{- 2} \times x\right) - \left(\textcolor{b l u e}{- 2} \times \textcolor{red}{4}\right)$

$y = - 2 x - \left(- 8\right)$

$y = \textcolor{red}{- 2} x + \textcolor{b l u e}{8}$