Equation of a Line Passing Through the Solution of a Pair of Linear Equations

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question: Write an equation of a line passing through the point representing the solution of the pair of linear equations

x + y = 2  …… (1)

2x − y = 1  …… (2)

How many such lines can we find?

Step 1: Find the Solution of the Given Pair of Equations

Add equations (1) and (2):

(x + y) + (2x − y) = 2 + 1

3x = 3

⇒ x = 1

Substitute x = 1 in equation (1):

1 + y = 2

⇒ y = 1

So, the solution point is (1, 1).

Step 2: Write the Equation of a Line Passing Through (1, 1)

The general equation of a line passing through (x₁, y₁) is:

y − y₁ = m(x − x₁)

Here, x₁ = 1 and y₁ = 1

⇒ y − 1 = m(x − 1)

This is the required equation of a line passing through the point (1, 1), where m is the slope of the line.

Step 3: Number of Such Lines

Since m can take infinitely many real values,

∴ infinitely many lines can be drawn passing through the point (1, 1).

Final Answer

The equation of a line passing through the solution point (1, 1) is:

y − 1 = m(x − 1)

where m is any real number.

Number of such lines = infinitely many.

Conclusion

Thus, the given pair of linear equations intersect at the point (1, 1), and infinitely many straight lines can be drawn passing through this point.