# How do you write an equation in slope intercept form for the line through the given points (7,5 ); (-1, 1/5)?

##### 2 Answers
Feb 17, 2017

$y = \textcolor{red}{\frac{3}{5}} x + \textcolor{b l u e}{\frac{4}{5}}$

#### Explanation:

First, we need to determine the slope. 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:

$m = \frac{\textcolor{red}{\frac{1}{5}} - \textcolor{b l u e}{5}}{\textcolor{red}{- 1} - \textcolor{b l u e}{7}} = \frac{\textcolor{red}{\frac{1}{5}} - \left(\frac{5}{5} \times \textcolor{b l u e}{5}\right)}{\textcolor{red}{- 1} - \textcolor{b l u e}{7}} = \frac{\textcolor{red}{\frac{1}{5}} - \frac{25}{5}}{\textcolor{red}{- 1} - \textcolor{b l u e}{7}}$

$m = \frac{- \frac{24}{5}}{-} 8 = \frac{24}{40} = \frac{8 \times 3}{8 \times 5} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{8}}} \times 3}{\textcolor{red}{\cancel{\textcolor{b l a c k}{8}}} \times 5} = \frac{3}{5}$

Now we can 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 first point from the problem gives:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x - \textcolor{red}{7}\right)$

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. Solving the equation we found for $y$ gives:

$y - \textcolor{red}{5} = \left(\textcolor{b l u e}{\frac{3}{5}} \times x\right) - \left(\textcolor{b l u e}{\frac{3}{5}} \times \textcolor{red}{7}\right)$

$y - \textcolor{red}{5} = \frac{3}{5} x - \frac{21}{5}$

$y - \textcolor{red}{5} + 5 = \frac{3}{5} x - \frac{21}{5} + 5$

$y - 0 = \frac{3}{5} x - \frac{21}{5} + \left(5 \times \frac{5}{5}\right)$

$y = \frac{3}{5} x - \frac{21}{5} + \frac{25}{5}$

$y = \textcolor{red}{\frac{3}{5}} x + \textcolor{b l u e}{\frac{4}{5}}$

Feb 17, 2017

$y = \frac{3}{5} x + \frac{4}{5}$

#### Explanation:

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = m x + b} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where m represents the slope and b, the y-intercept.

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

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

The 2 coordinate points are  (7,5)" and (-1,1/5)

let $\left({x}_{1} , {y}_{1}\right) = \left(- 1 , \frac{1}{5}\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(7 , 5\right)$

$\Rightarrow m = \frac{5 - \frac{1}{5}}{7 + 1} = \frac{\frac{24}{5}}{8} = \frac{3}{5}$

We can write the partial equation as $y = \frac{3}{5} x + b$

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

$\text{Using " (7,5)" that is } x = 7 , y = 5$

$\Rightarrow 5 = \left(\frac{3}{5} \times 7\right) + b$

$\Rightarrow b = 5 - \frac{21}{5} = \frac{25}{25} - \frac{21}{25} = \frac{4}{5}$

$\Rightarrow y = \frac{3}{5} x + \frac{4}{5} \text{ is equation in slope-intercept form}$