# How do you find slope, point slope, slope intercept, standard form, domain and range of a line for Line I (3,4) (0,10)?

Jun 5, 2017

Slope: $- 2$

Point-Slope Equation: $y - 10 = - 2 x$
Slope-Intercept Equation: $y = - 2 x + 10$
Standard Form Equation: $2 x + y = 10$

Domain: $x \in \mathbb{R}$
Range: $y \in \mathbb{R}$

#### Explanation:

Let's start with domain and range. Think about drawing a straight line on a coordinate plane. As long as the line is not horizontal or vertical, you should be able to choose any $x$ or $y$ value and find a point on the line with that value.

In simpler terms, no value of $x$ will make $y$ undefined for a linear equation. So, the domain is all real numbers, or $x \in \mathbb{R}$.

In simpler terms, any non-horizontal line stretches from the very bottom of the Cartesian plane ($- \infty$) to the very top ($\infty$), so the range (or all possible $y$ values) is all real numbers, or $y \in \mathbb{R}$.

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Now, let's find the slope of our line. Everything else will follow:

$\text{slope" = "change in y"/"change in x} = \frac{10 - 4}{0 - 3} = \frac{6}{-} 3 = - 2$

The point-slope equation for a line looks like this:

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

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

Using $\left(0 , 10\right)$ as our point and $- 2$ as our slope, the equation is:

$y - 10 = - 2 \left(x - 0\right)$

Or:

$y - 10 = - 2 x$

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Now to find slope-intercept. We are looking for something that looks like this:

$y = m x + b$

Where $m$ is the slope and $\left(0 , b\right)$ is the $y$ intercept. Luckily, one of the points we were given (0,10) IS the $y$ intercept, since its $x$ value is 0. Therefore, $b = 10$, and we already know $m = - 2$. So:

$y = - 2 x + 10$

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Finally, we need to find the standard form equation. It will look like this:

$a x + b y = c$

Where $a$, $b$, and $c$ are integers with a GCF of 1, and $a$ is positive.

To find the standard form, let's take our slope-intercept form and manipulate it until we get something that looks like standard form:

$y = - 2 x + 10$

$y + 2 x = - 2 x + 10 + 2 x$

$2 x + y = 10$

This meets all the requirements for standard form, so it is our final equation!