Question

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

Answer 1

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 $y = x + 2$