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.

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