Answer

Verified

398.4k+ views

**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.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

Trending doubts

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

10 examples of evaporation in daily life with explanations