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Question: How do you find the equation of line passing through the point \(P\left( 8,2 \right)\) with a slope ...

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

Explanation

Solution

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
(y−y1)=m(x−x1)\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)
Where, m=m= slope of the line
(x1,y1)=\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+by=mx+b
Where, m=m= slope of the line
b=b= yy-intercept.

Complete step by step solution:
Given that, there is point P(8,2)P\left( 8,2 \right) from which a line is passing whose slope is 4.4.
We know the formula of point slope, from that we can find out the equation.
So, the formula for point-slope formula is,
(y−y1)=m(x−x1)...(i)\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)...(i)
Where, mm is slope is given in the numerical i.e. 44 and (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) are the points from which the line is passing i.e. (8,2)\left( 8,2 \right)
From above we can say that,
m=4m=4
x1=8{{x}_{1}}=8
y1=2{{y}_{1}}=2
Put this value in equation (i)(i) we get,
(y−2)=4(x−8)...(ii)\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)y=mx+b...(iii)
Now, simplify the equation (ii)(ii)
(y−2)=4(x−8)\left( y-2 \right)=4\left( x-8 \right)
Multiply whole bracket with 44 on right side
y−2=4×x−4×8y-2=4\times x-4\times 8
⇒y−2=4x−32\Rightarrow y-2=4x-32
Transpose 22 on the right side of the equation,
y=4x−32+2y=4x-32+2
⇒y=4x−30...(iv)\Rightarrow y=4x-30...(iv)
Now compare equation (iii)(iii) and (iv)(iv)
y=mx+by=mx+b
⇒y=4x−30\Rightarrow y=4x-30
From above, the value of m=4m=4 and b=−30b=-30

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

Additional Information:
The formula for slope intercept for a linear equation is given as,
y=mx+by=mx+b
Where, m=m=slope of the line
b=yb=y-intercept
The slope (m)(m) in above formula is nothing but eh change in yy to the change in xx
It is generally called as ratio of rise to run i.e. riserun\dfrac{rise}{run}
It can also be written as ΔyΔx\dfrac{\Delta y}{\Delta x}. When we have two point on the line 9.9. (x1,y1)&(x2,y2)\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right) then we can use following formula for calculating slope (m)(m)
The formula of slope for 22 points.
m=y2−y1x2−x1m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
The yy-intercept is nothing but the value where the yy-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)(m) But the value of mm is already given. So we have to calculate the value of bb 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.