Equation of a Line Passing Through a Given Point

Video Explanation

Question

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

\[ x + y = 2, \quad 2x – y = 1 \]

How many such lines can we find?

Solution

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

Add the two equations:

\[ (x + y) + (2x – y) = 2 + 1 \]

\[ 3x = 3 \]

\[ x = 1 \]

Substitute \(x = 1\) in \(x + y = 2\):

\[ 1 + y = 2 \Rightarrow y = 1 \]

So, the solution point is:

\[ (1,\;1) \]

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

The general form of a line passing through a point \((x_1, y_1)\) is:

\[ y – y_1 = m(x – x_1) \]

Here, \[ (x_1, y_1) = (1, 1) \]

So, the equation becomes:

\[ y – 1 = m(x – 1) \]

where \(m\) is any real number.

Step 3: Answer the Question

Since \(m\) can take infinitely many real values, we can write infinitely many lines passing through the point \((1, 1)\).

Conclusion

An equation of a line passing through the solution point is:

\[ y – 1 = m(x – 1) \]

The number of such lines is:

\[ \boxed{\text{Infinitely many}} \]

\[ \therefore \quad \text{Infinitely many lines can be drawn through the given point.} \]