# What is the slope-intercept form of the equation which goes through (1, 1) and (3, 4)?

Jul 25, 2017

See a solution process below:

#### 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:

$m = \frac{\textcolor{red}{4} - \textcolor{b l u e}{- 1}}{\textcolor{red}{3} - \textcolor{b l u e}{- 1}} = \frac{\textcolor{red}{4} + \textcolor{b l u e}{1}}{\textcolor{red}{3} + \textcolor{b l u e}{1}} = \frac{5}{4}$

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.

We can substitute the slope for we calculated $m$ and the values from one for the points for $x$ and $y$ and solve for $b$:

$y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$ becomes:

$4 = \left(\textcolor{red}{\frac{5}{4}} \cdot 3\right) + \textcolor{b l u e}{b}$

$4 = \textcolor{red}{\frac{15}{4}} + \textcolor{b l u e}{b}$

$4 - \frac{15}{4} = \textcolor{red}{\frac{15}{4}} - \frac{15}{4} + \textcolor{b l u e}{b}$

$\left(\frac{4}{4} \cdot 4\right) - \frac{15}{4} = 0 + \textcolor{b l u e}{b}$

$\frac{16}{4} - \frac{15}{4} = \textcolor{b l u e}{b}$

$\frac{1}{4} = \textcolor{b l u e}{b}$

We can now substitute the slope and value for $b$ we calculated into the formula to write the equation:

$y = \textcolor{red}{\frac{5}{4}} x + \textcolor{b l u e}{\frac{1}{4}}$