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How do you find the equation of line passing through the point $P\left( 8,2 \right)$ with a slope of $4?$

Last updated date: 22nd Jul 2024
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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.
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,
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,
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,
$\Rightarrow y=4x-30...(iv)$
Now compare equation $(iii)$ and $(iv)$
$\Rightarrow y=4x-30$
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,
Where, $m=$slope of the line
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.
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.