# How do you find slope, point slope, slope intercept, standard form, domain and range of a line for Line F (6,0) (10,3)?

Jul 27, 2017

See a solution process below:

#### Explanation:

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:

$m = \frac{\textcolor{red}{3} - \textcolor{b l u e}{0}}{\textcolor{red}{10} - \textcolor{b l u e}{6}} = \frac{3}{4}$

Point Slope

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 $\left(\textcolor{red}{{x}_{1} , {y}_{1}}\right)$ is a point the line passes through.

Substituting the slope we calculated and the values from the first point gives:

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{\frac{3}{4}} \left(x - \textcolor{red}{6}\right)$

We can also substitute the slope we calculated and the values from the second point giving:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{\frac{3}{4}} \left(x - \textcolor{red}{10}\right)$

Slope Intercept

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 take the first point-slope equation and solve for $y$:

$y - \textcolor{red}{0} = \textcolor{b l u e}{\frac{3}{4}} \left(x - \textcolor{red}{6}\right)$

$y = \left(\textcolor{b l u e}{\frac{3}{4}} \times x\right) - \left(\textcolor{b l u e}{\frac{3}{4}} \times \textcolor{red}{6}\right)$

$y = \frac{3}{4} x - \frac{18}{4}$

$y = \textcolor{red}{\frac{3}{4}} x - \textcolor{b l u e}{\frac{9}{2}}$

Standard Form

The standard form of a linear equation is: $\textcolor{red}{A} x + \textcolor{b l u e}{B} y = \textcolor{g r e e n}{C}$

Where, if at all possible, $\textcolor{red}{A}$, $\textcolor{b l u e}{B}$, and $\textcolor{g r e e n}{C}$are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

We can convert the slope intercept to Standard Form as follows:

$y = \textcolor{red}{\frac{3}{4}} x - \textcolor{b l u e}{\frac{9}{2}}$

$- \frac{3}{4} x + y = - \frac{3}{4} x + \textcolor{red}{\frac{3}{4}} x - \textcolor{b l u e}{\frac{9}{2}}$

$- \frac{3}{4} x + y = 0 - \textcolor{b l u e}{\frac{9}{2}}$

$- \frac{3}{4} x + y = - \textcolor{b l u e}{\frac{9}{2}}$

$\textcolor{red}{- 4} \left(- \frac{3}{4} x + y\right) = \textcolor{red}{- 4} \times - \textcolor{b l u e}{\frac{9}{2}}$

$3 x - 4 y = \frac{36}{2}$

$\textcolor{red}{3} x + \textcolor{b l u e}{- 4} y = \textcolor{g r e e n}{18}$

Domain and Range

There are no restrictions on the value of $x$, therefore the Domain is: The set of of Real Numbers or $\left\{\mathbb{R}\right\}$

From the slope-intercept form, as $x$ increases, $y$ increases, as $x$ decreases, $y$ decreases. Therefore the Range is: The set of of Real Numbers or $\left\{\mathbb{R}\right\}$