Use the slope formula: 

$\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{-2-4}{8-5} = \frac{-6}{3} = -2$

To calculate the slope of a line, we pick two points, and calculate the change in $\displaystyle y$ divided by the change in $\displaystyle x$. To calculate the change in value, we use subtraction. So, 

slope = $\displaystyle \frac{\text{Change in Y}}{\text{Change in X}} = \frac{y_2-y_1}{x_2-x_1}$ 

So we can pick any two points and plug them into this formula. Let's choose (0,3) and (4,6). We get 

$\displaystyle \frac{6-3}{4-0}= \frac{3}{4}$

The general formula of a straight line is $\displaystyle y=mx+b$, where $\displaystyle \small m$ is the slope and $\displaystyle \small b$ is the y-intercept.

To evaluate our original equation, we can compare it to this formula.

$\displaystyle y=mx+b$

$\displaystyle y=-2x-3$

The slope is $\displaystyle \small -2$ and the y-intercept is at $\displaystyle \small -3$. Since the y-intercept is a point on the line, it is written as $\displaystyle \small (0,-3)$.

Rewrite in slope-intercept form:

$\displaystyle 2x + 6y = 15$

$\displaystyle -2x + 2x + 6y = -2x +15$

$\displaystyle 6y = -2x +15$

$\displaystyle 6y \div 6= \left (-2x +15 \right )\div 6$

$\displaystyle y= - \frac{1}{3}x +\frac{5}{2}$

The slope is the coefficient of $\displaystyle x$:

 $\displaystyle m = - \frac{1}{3}$

This function describes a straight line. We can compare the given equation with the formula for a straight line in slope-intercept form.

$\displaystyle y=-\frac{1}{3}x-2$

$\displaystyle y=mx+b$

In the formula, $\displaystyle m$ is the slope and $\displaystyle b$ is the y-intercept. Looking at our given equation, we can see that $\displaystyle m=-\frac{1}{3}$ and $\displaystyle b=-2$.

Since the y-intercept is a point, we want to write it in point notation: $\displaystyle (0,-2)$

$\displaystyle 1-10x+5y=2$

You see that this equation is not written explicitly in terms of $\displaystyle x$. Before we can determine the slope and y-intercept, we need to write the equation explicitly in terms of $\displaystyle x$ by solving for $\displaystyle y$.

First, add $\displaystyle 10x$ to both sides.

$\displaystyle 1-10x+10x+5y=10x+2$

$\displaystyle 1+5y=10x+2$

Then subtract $\displaystyle 1$ to both sides.

$\displaystyle 1-1+5y=10x+2-1$

$\displaystyle 5y=10x+1$

Finally, divide by $\displaystyle 5$.

$\displaystyle \frac{5y}{5}=\frac{10x+1}{5}$

$\displaystyle y=2x+\frac{1}{5}$

Now that the equation is explicitly in terms of $\displaystyle x$, we can compare it to the general formula of a straight line in slope-intercept form.

$\displaystyle y=mx+b$

In this equation, $\displaystyle m$ is equal to the slope and $\displaystyle b$ is equal to the y-intercept. Comparing our given equation to the formula, we can see that $\displaystyle m=2$ and $\displaystyle b=\frac{1}{5}$.

Since the y-intercept is a point, we will want to write it in point notation: $\displaystyle (0,\frac{1}{5})$.

To determine the slope, we use the following formula.

$\displaystyle m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

In the formula, the two points making up that slope are $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$.  In our case, our two points are $\displaystyle (-2,4)$ and $\displaystyle (3,-1)$. Using these values in the formula allows us to solve for the slope.

$\displaystyle m=\frac{-1-4}{3-(-2)}=\frac{-5}{5}=-1$

The slope is $\displaystyle -1$.

To find the slope of the line passing through the given points, we need to find the change in y and divide it by the change in x.  This can also be written as $\displaystyle \frac{y_{1}-y_0}{x_1-x_0}$.  In our case, we have $\displaystyle \frac{-2-2}{5-0}=\frac{-4}{5}=-0.80$

To find the slope, divide the change in $\displaystyle y$ by the change in $\displaystyle x$.

So,

 $\displaystyle \frac{6-5}{5-2}=\frac{1}{3}$

Check your answer by using the same equation on another pair of points:

$\displaystyle \frac{7-6}{8-5}=\frac{1}{3}$

It's the same! So the answer is $\displaystyle \frac{1}{3}$.

To easily determine an equation's y-intercept, convert it to the $\displaystyle y=mx+b$ form, where the $\displaystyle b$ represents the equation's y-intercept.

Converting the given equation to this form gives you 

$\displaystyle y=\frac{1}{2}x+\frac{3}{2}$ 

with a y-intercept of $\displaystyle \frac{3}{2}$.