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

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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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 \[\dfrac{{\left( {y - {y_1}} \right)}}{{\left( {x - {x_1}} \right)}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\].

Complete step-by-step answer:
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.