Question

How do you find an equation of the straight line passing through the points with coordinates \[(-1,5)\] and \[(4,-2)\]. Giving your answer in the form of \[ax+by+c=0\]?

Answer 1

Hint: Here, we are required to find the equation of a line passing through two given points. We will use the formula of the equation of a line which passes through the points \[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},{{y}_{2}})\]. We will then substitute the given points to find the required equation.

Complete step-by-step solution:
To solve his type of problem, we need to find the equation of line of pass through two points for that
We need to find the slope of line which is given by formula that is given by
\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
For better understanding of the question figure is given below:

In this above figure, it has mentioned the line AB. We have to find the equation of line AB in such a way that the line passes through two points that are with coordinates \[(-1,5)\] and \[(4,-2)\].
So, first of all we need to find the slope. When you notice this coordinates in the figure then you can see that \[{{x}_{1}}=-1\], \[{{x}_{2}}=4\], \[{{y}_{1}}=5\] and \[{{y}_{2}}=-2\]
Formula is given by \[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
Substitute the values in the above formula we get;
 \[\Rightarrow m=\dfrac{-2-5}{4-\left( -1 \right)}\]
By simplifying this we get:
\[\Rightarrow m=\dfrac{-7}{5}----(1)\]
Now, we have to find the equation of a line using a given point and slope, we use the formula that is
\[\Rightarrow y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)\]
Or we can write this as:
\[\Rightarrow m=\dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}---(2)\]
Substitute the value of equation (1) in equation (2) we get:
\[\Rightarrow \dfrac{-7}{5}=\dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}\]
Hence the substituting the value of \[{{x}_{1}}=-1\] and \[{{y}_{1}}=5\] on above equation we get:
\[\Rightarrow \dfrac{-7}{5}=\dfrac{y-5}{x-(-1)}\]
By simplifying this we get:
\[\Rightarrow -7\left( x-(-1) \right)=5\left( y-5 \right)\]
By solving this bracket, we get:
\[\Rightarrow -7x-7=5y-25\]
Now, according to the question we have to write in the form of \[ax+by+c=0\]
To get in the form of equation, first of all we need to rearrange the term then we get:
 \[\Rightarrow -7x-5y+18=0\]
In above equation multiply minus on both sides we get:
\[\Rightarrow 7x+5y-18=0\].
Hence, the equation of the line passing through the points \[(-1,5)\] and \[(4,-2)\]

Note: In the standard form, an equation of a straight line is written as \[y=mx+c\]. Here \[m\] is the slope. A slope of a line states how steep a line is and in which direction the line is going.
When we are required to find an equation of a given line then, we use the relation between \[x\] and \[y\] coordinates of any point present on that specific line to find its equation. Another approach to solve this problem by using the method of two-point form to get the required equation of line. You can also use direct straight-line equations to solve this type of problem.