# How do you find slope, point slope, slope intercept, standard form, domain and range of a line for Line B (2, -7) (0, -10)?

Oct 22, 2017

See a solution process below:

#### Explanation:

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:

$m = \frac{\textcolor{red}{- 10} - \textcolor{b l u e}{- 7}}{\textcolor{red}{0} - \textcolor{b l u e}{2}} = \frac{\textcolor{red}{- 10} + \textcolor{b l u e}{7}}{\textcolor{red}{0} - \textcolor{b l u e}{2}} = \frac{- 3}{-} 2 = \frac{3}{2}$

We know from the problem the $y$ intercept is $\left(0 , - 10\right)$ or $- 10$

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.

Substituting the slope we calculated and the $y$-intercept gives:

$y = \textcolor{red}{\frac{3}{2}} x - \textcolor{b l u e}{10}$

The point-slope form of a linear equation is: $\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

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

We can substitute the slope we calculated and the values from the first point in the problem giving:

$\left(y - \textcolor{b l u e}{- 7}\right) = \textcolor{red}{\frac{3}{2}} \left(x - \textcolor{b l u e}{2}\right)$

$\left(y + \textcolor{b l u e}{7}\right) = \textcolor{red}{\frac{3}{2}} \left(x - \textcolor{b l u e}{2}\right)$

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

$\left(y - \textcolor{b l u e}{- 10}\right) = \textcolor{red}{\frac{3}{2}} \left(x - \textcolor{b l u e}{0}\right)$

$\left(y + \textcolor{b l u e}{10}\right) = \textcolor{red}{\frac{3}{2}} x$

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 take this last point slope equation and convert it to standard form as follows:

$2 \left(y + \textcolor{b l u e}{10}\right) = 2 \times \textcolor{red}{\frac{3}{2}} x$

$2 y + 20 = 3 x$

$2 y - 2 y + 20 = 3 x - 2 y$

$0 + 20 = 3 x - 2 y$

$20 = 3 x - 2 y$

$3 x - 2 y = 20$

The are no constraints on the value $x$ can take in this equation therefore the Domain is the set of All Real Numbers or $\left\{\mathbb{R}\right\}$

Because this is a pure linear transformation the Range of this equation is also the set of All Real Numbers or $\left\{\mathbb{R}\right\}$