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Question: How do you write an equation in point slope form given (-1,3) and (1,7)?...

How do you write an equation in point slope form given (-1,3) and (1,7)?

Explanation

Solution

This question is from the topic of straight line. In this question, we have to find the equation of line using the given points. In solving the question, we will first find out the slope of the line using the given points. After that, we will understand about the point-slope form of the straight line. After that, we will use the found slope and any one of the given points, we will find the equation of line.

Complete step by step answer:
Let us solve this question.
In this question, we have given two points. From these two points, we will find out the equation in point slope form.
Let us first find out the slope of the line using these points.
If the points are (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and (x2,y2)\left( {{x}_{2}},{{y}_{2}} \right), then the slope of line is m=y2−y1x2−x1m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.
So, we can write the slope of line as
m=7−31−(−1)=42=2m=\dfrac{7-3}{1-\left( -1 \right)}=\dfrac{4}{2}=2
Now, let us understand about point-slope form.
If the point on a line is (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and the slope of the line is m, the point-slope form of a line can be written as
y−y1=m(x−x1)y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)
So, we can write the equation of the line using any of the points that is (-1,3) and (1,7) and slope as m=2.
Let us use the point here (1,7) .
The equation of line in point-slope form will be
y−7=2(x−1)y-7=2\left( x-1 \right)
The above equation can also be written as
⇒y−7=2x−2\Rightarrow y-7=2x-2
⇒y=2x−2+7\Rightarrow y=2x-2+7
The above equation can also be written as
⇒y=2x+5\Rightarrow y=2x+5
Hence, we have found the equation using point slope form. The equation in point slope form is y−7=2(x−1)y-7=2\left( x-1 \right) and we have solved this equation as y=2x+5y=2x+5.
We can take reference from the following figure:


Note:
We should have a proper knowledge in the topic of straight line to solve this type of question easily. We should know how to find out the slope from the two given points. If the points are given as (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and (x2,y2)\left( {{x}_{2}},{{y}_{2}} \right), then the slope of line will be m=y2−y1x2−x1m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}. And, we should know about the point slope form of a line. If a point is given as (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and slope is given as m, then equation of line in point-slope form will be:
y−y1=m(x−x1)y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)