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# The equation of the line passing through the points $\left( {2,3} \right)$ and $\left( {4,5} \right)$ isA. $x - y - 1 = 0$B. $x + y + 1 = 0$C. $x + y - 1 = 0$D. $x - y + 1 = 0$

Last updated date: 09th Aug 2024
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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 $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$. We will then substitute the given points to find the required equation.

Formula Used:
Equation of a line which passes through 2 points is given by $\dfrac{{\left( {y - {y_1}} \right)}}{{\left( {x - {x_1}} \right)}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.

When we have to find the equation of a line using a given point and slope, we use the formula $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$.
Or we can write this as:
$m = \dfrac{{\left( {y - {y_1}} \right)}}{{\left( {x - {x_1}} \right)}}$………………………………(1)
Also, slope of a given line which passes through the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is:
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Putting $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ value in equation (1), we get,
$\Rightarrow \dfrac{{\left( {y - {y_1}} \right)}}{{\left( {x - {x_1}} \right)}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Hence, this is the formula for the equation of a line which passes through the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$.
Now, according to the question, we have to find the equation of the line passing through the points $\left( {2,3} \right)$ and $\left( {4,5} \right)$.
Hence, substituting ${x_1} = 2$, ${y_1} = 3$ and ${x_2} = 4$,${y_2} = 5$ in the formula $\dfrac{{\left( {y - {y_1}} \right)}}{{\left( {x - {x_1}} \right)}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$, we get

$\dfrac{{\left( {y - 3} \right)}}{{\left( {x - 2} \right)}} = \dfrac{{5 - 3}}{{4 - 2}}$
Subtracting the terms, we get
$\Rightarrow \dfrac{{\left( {y - 3} \right)}}{{\left( {x - 2} \right)}} = \dfrac{2}{2} = \dfrac{1}{1}$
Now, by cross multiplying the terms, we get
$\Rightarrow \left( {y - 3} \right) = \left( {x - 2} \right)$
Now, subtracting $\left( {y - 3} \right)$ from both sides, we get
$\Rightarrow 0 = x - 2 - y + 3$
$\Rightarrow 0 = x - y + 1$
Or
$\Rightarrow x - y + 1 = 0$
Hence, the equation of the line passing through the points $\left( {2,3} \right)$ and $\left( {4,5} \right)$ is $x - y + 1 = 0$

Therefore, option D is the correct answer.

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.