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Find the equations of line joining points (−2,3) and (1,4)?

Find the equations of line joining points (−2,3) and (1,4)?

Answer 1:
The equation for line joining two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by \(\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\) and for given points it is \(y=\frac{1}{3} x+\frac{11}{3}\)

Explanation:
Let the slope intercept form of equation be \(y=m x+c\)
here we do not know the slope \(m\) and \(y-\) intercept \(C\)
What we know is that this passes through the two coordinate pairs, say \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\).

As such we have three equations

\(y=m x+c \ldots \ldots(1)\)
\(y_{1}=m x_{1}+c \ldots \ldots(2) \text { and }\)
\(y_{2}=m x_{2}+c \ldots \ldots(3)\)

Now using these let us eliminate \(m\) and \(c\)

subtracting (2) from (1), we get \(\left(y-y_{1}\right)=m\left(x-x_{1}\right) \ldots \ldots(4)\)
and subtracting (2) from (3), we get \(\left(y-2-y_{1}\right)=m\left(x_{2}-x_{1}\right) \ldots \ldots(5)\)

Dividing (4) by (5)

\(\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\)

As the two points are \((-2,3)\) and \((1,4)\), the equation is
Find the equations of line joining points (−2,3) and (1,4) Img 1

Answer 2:
For any point on the line, the coordinate pair in slope-intercept form is

\(\left(x, \frac{x}{3}+\frac{11}{3}\right)\)

Explanation:
For slope m and intercept c, the equation is y = m x +c.

The slope intercept form for coordinates is (x, m x +c ).

The slope of the line through the given points is \(m=\frac{4-3}{1-(-2)}=\frac{1}{3}\)

Also, from (1, 4). 4 = i/3(1) + c. So, c = 11/3.

So, the answer is \(\left(x, \frac{1}{3} x+\frac{11}{3}\right)\).

Answer 3:
The equation of the line is:

\(y=\frac{1}{3} x+\frac{11}{3}\) which can be written as \(y=\frac{1}{3} x+3 \frac{2}{3}\)

Explanation:
If you are given the coordinates of 2 points on a line, substituting them into the formula below allows you to find the equation immediately. In the process you also calculate the slope.
Find the equations of line joining points (−2,3) and (1,4) Img 2

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