Question

What is the equation of the line passing through \[(1,\ 3)\] and \[(4,\ 6)\] ?

Answer 1

Hint: In this question, we need to find the equation of the line passing through \[(1,\ 3)\] and \[(4,\ 6)\] . First, by using the gradient formula, we need to find the slope of the line. It is calculated by dividing the change in \[y\] coordinate by change in \[x\] co-ordinate. Mathematically, slope is denoted by the letter \[m\] . First, by using the slope formula, we can find the slope of the line. Then by using slope intercept form, we can find the equation of the line.
Formula used :
Slope of the line is
\[Slope\ = \dfrac{\left( \text{change in y} \right)}{\text{change in slope}}\]
\[\Rightarrow \ m = \dfrac{\left( y_{2} – y_{1} \right)}{x_{2} – x_{1}}\]
Where \[m\] is the slope .
Slope intercept form :
The equation of line is
\[y = mx + c\]
Where \[m\] is the slope and \[c\] is the y-intercept.

Complete step by step answer:
Given, two points \[(1,\ 3)\] and \[(4,\ 6)\]
Let us consider \[\left( x_{1},\ y_{1} \right)\] be \[(1,\ 3)\] and \[\left( x_{2},\ y_{2} \right)\] be \[(4,\ 6)\] .
By using slope formulas, we can find the slope .
\[\ m = \dfrac{\left( y_{2} - y_{1} \right)}{x_{2} - x_{1}}\]
By substituting the values we get,
\[\Rightarrow \ m = \dfrac{\left( 6 - 3 \right)}{4 - 1}\]
On simplifying we get,
\[m = \dfrac{3}{3} = 1\]
We can substitute the value of \[m\] in the slope intercept form to find
the value of y-intercept.
Slope intercept form is \[y = mx + c\]
By substituting the value of \[m\], We get
\[y = x + c\] which is known as the partial equation.
In order to find the find the value of \[c\] in the partial equation ,
we need to substitute any one of the given points in the partial
equation.
Now we can substitute the point \[(1,\ 3)\] in the partial equation.
Here \[x = 1\] and \[y = 3\]
Thus we get,
\[\ 3 = 1 + c\]
\[\Rightarrow \ c = 3 - 1\]
On simplifying we get,
\[c = 2\]
Thus again by substituting the value of \[c\] in the slope intercept
form, we get the equation of the line.
When \[c = 2\] , the equation of the line is \[y = x + 2\]
Thus we get the equation of the line is \[y = x + 2\]

Note:
Alternative solution :
There is a direct formula to find the equation of the line passing through two points.
Formula used :
The equation of the a lines passing through the points \[\left( x_{1},\ y_{1} \right)\] and \[\left( x_{2},\ y_{2} \right)\] is
\[\dfrac{y – y_{1}}{y_{2} – y_{1}} = \dfrac{x – x_{1}}{x_{2} – x_{1}}\]
Let us consider \[\left( x_{1},y_{1} \right)\] be \[(1,3)\] and \[\left( x_{2},y_{2} \right)\] be \[(4,6)\]
On substituting the values in the formula,
We get,
\[\dfrac{y – 3}{6 – 3} = \dfrac{\left( x – 1 \right)}{4 – 1}\]
On simplifying,
We get,
\[\dfrac{y – 3}{3} = \dfrac{x – 1}{3}\]
On cancelling the denominator,
We get,
\[(y – 3)\ = (x – 1)\]
\[\Rightarrow \ y = x + 3 – 1\]
On simplifying,
We get,
\[y = x + 2\]
The equation of the line is \[y = x + 2\]