Finding a Linear Equation

We substitute into the linear equation $y=mx+b$
The given slope $m$ and the values of the given point. This is how we will find $b$ and can determine the linear equation.

Given a point through which the line passes: $\left(2,4\right)$ and the slope: $-2$
Find the linear equation.

Solution:

We substitute the slope and the given point into the linear equation:
$4=-2\times2+b$

We obtain:
$4=-4+b$
To find $b$  
$b=8$

Now, we have both the slope given in the question and $b$.
We can determine that the linear equation is:
$y=-2x+8$

In this way, we'll first find the slope using $2$ points according to the formula.
After that, we'll find the linear equation using the first form (by the slope and a point)
The formula to find the slope using $2$ points is:

$m=\frac{\left(Y2-Y1\right)}{\left(X2-X1\right)}$

Given the following two points through which the line passes:
$\left(3,7\right),\left(6,1\right)$
Find the linear equation.

Solution:
First, find the slope using the formula. Replace the given points and you will get:

$m=\frac{1-7}{6-3}$

$m=\frac{-6}{3}$

$m=-2$

Now that we have found the slope, we can use the point-slope form. We will choose a point from the given ones and place the slope and the chosen point into the linear equation template.

We obtain:

$7=-2\times3+b$

$7=-6+b$

$b=13$

Now, we have both the slope we found and $b$, and we can determine that the linear equation is:
$y=-2x+13$

When you are given a parallel line to another line you are looking for, you should know that the slope of the parallel line is the same as the slope of the line you are looking for. Therefore, you can take the slope of the parallel line and assume it is the slope of the line you are seeking. Pay attention: you can identify the slope only in an explicit equation where Y is isolated, stands alone on one side of the equation, and its coefficient is $1$.

Usually, you will be given a point and then you can find the linear equation using a point and a slope
(the point-slope form).

Find the linear equation that passes through the point: $\left(6,5\right)$ and is parallel to the line $y=3x-7$

Solution:

We are given that the line is parallel to the line $y=3x-7$.
This information tells us that the slope of our line is the same as the slope of the corresponding line, and therefore the slope of the equation of the line we are looking for is $3$.
Now, we have the slope $3$ and the point $5,6$.
By substituting into the linear equation formula, we will find $b$ and thus we will find the linear equation (the first form).

Pay attention!
We easily extracted the slope since the equation of the parallel line is given explicitly with $Y$ on one side of the equation and its coefficient is $1$.
If we are given an equation that is not in the explicit form, like this one for example: $5=3y+6$
We need to get to an explicit equation: to isolate $Y$ completely and only then identify the slope.

The product of the slopes of perpendicular lines is$-1$.
Therefore, when we are given a line perpendicular to the line we are looking for, we know that the product of the two slopes is $-1$ and this is how we will find the slope.

The requested line is perpendicular to the line $y=2x-6$
What is the slope of the required line?

Solution:
Define the slope of the requested line as $m$.
Since both lines are perpendicular, we can derive the slope of the perpendicular line $-2$
and write the equation where the product of the two slopes is equal to $-1$:

We obtain:

$2\times m=-1$

Find $m$:
$m=-0.5$

The slope of the required line is $-0.5$ .
Now, we will find the equation of the line based on the slope and the given point.

Note:

Here too, it is important to first observe what the equation of the perpendicular line expresses.

When you're given the graph of a function, you can find the linear equation.
First, choose $2$ points on the graph.
This way, you'll find the slope of the function (according to the second method).
Then, find the linear equation based on any point you choose, which the straight line passes through, and of course, the slope you found (the first method).

Given two pairs of lines:

$Y=3X+2$

$Y=3X-5$

$Y=2X-6$

$Y=-0.5X+9$

The first pair is a pair of parallel lines because the slopes $m1=m2=3$ are equal.

The second pair is a pair of perpendicular lines because their slopes satisfy $2\times-0.5=-1$. That is, $m1\times m2=-1$