Answer

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**Hint:**Here, the points are given with its slope and we have to make the equation.

For that we have to first use the point-slope formula as the points and slope are given.

The formula for points slope is

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

Where, $m=$ slope of the line

$\left( {{x}_{1}},{{y}_{1}} \right)=$ Points given in the numerical from this formula we can calculate the equation. After simplifying the equation, compare that equation with standard slope intercept formula i.e.

$y=mx+b$

Where, $m=$ slope of the line

$b=$ $y$-intercept.

**Complete step by step solution:**

Given that, there is point $P\left( 8,2 \right)$ from which a line is passing whose slope is $4.$

We know the formula of point slope, from that we can find out the equation.

So, the formula for point-slope formula is,

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

Where, $m$ is slope is given in the numerical i.e. $4$ and $\left( {{x}_{1}},{{y}_{1}} \right)$ are the points from which the line is passing i.e. $\left( 8,2 \right)$

From above we can say that,

$m=4$

${{x}_{1}}=8$

${{y}_{1}}=2$

Put this value in equation $(i)$ we get,

$\left( y-2 \right)=4\left( x-8 \right)...(ii)$

We have to convert the above equation in the form of slope-intercept equation.

The formula of the slope-intercept for the linear equation is as,

$y=mx+b...(iii)$

Now, simplify the equation $(ii)$

$\left( y-2 \right)=4\left( x-8 \right)$

Multiply whole bracket with $4$ on right side

$y-2=4\times x-4\times 8$

$\Rightarrow y-2=4x-32$

Transpose $2$ on the right side of the equation,

$y=4x-32+2$

$\Rightarrow y=4x-30...(iv)$

Now compare equation $(iii)$ and $(iv)$

$y=mx+b$

$\Rightarrow y=4x-30$

From above, the value of $m=4$ and $b=-30$

**Therefore the equation for the given point $P\left( 8,-2 \right)$ is $y=4x-30$**

**Additional Information:**

The formula for slope intercept for a linear equation is given as,

$y=mx+b$

Where, $m=$slope of the line

$b=y$-intercept

The slope $(m)$ in above formula is nothing but eh change in $y$ to the change in $x$

It is generally called as ratio of rise to run i.e. $\dfrac{rise}{run}$

It can also be written as $\dfrac{\Delta y}{\Delta x}$. When we have two point on the line $9.$ $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ then we can use following formula for calculating slope $(m)$

The formula of slope for $2$ points.

$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$

The $y$-intercept is nothing but the value where the $y$-axis is touched or intersects by the line.

**Note:**In this numerical the two equations are used. Both the equations are related to the slope-intercept equation.

The first formula used is slope-points which is used for nothing but calculating the slope $(m)$ But the value of $m$ is already given. So we have to calculate the value of $b$ for the slope intercept. Formula which is the second equation.

And we have to simplify that equation into slope intercept form.

We use the point-slope equation formula for getting the equation in the term of slope intercept which is the standard equation.

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