write an equation of a line passing through the point representing solution of the pair of linear equations 2x+3y=6 and 4x+6y=12 how many such lines can be find

To find the equation of a line passing through the solution of the pair of linear equations, we first need to find the point of intersection (if any) of the given lines. The given pair of linear equations are:

1. \( 2x + 3y = 6 \)
2. \( 4x + 6y = 12 \)

### Step 1: Analyze the given system of equations

Notice that the second equation is simply a multiple of the first:

\[
4x + 6y = 2(2x + 3y)
\]

This indicates that both equations represent the same line. Therefore, there are infinitely many solutions, as both lines coincide.

### Step 2: General equation of a line

Since the solution lies on this line, any line that passes through any point on the line \( 2x + 3y = 6 \) will satisfy the condition. The equation of a line passing through any point on this line can be written in the form:

\[
ax + by = c
\]

### Step 3: Conclusion

Since there are infinitely many points on the line \( 2x + 3y = 6 \), there are **infinitely many lines** that can be drawn through any point on this line. Therefore, infinitely many such lines can be found.

Would you like more details on how to create an explicit equation for any such line, or do you have any specific questions?

Here are some related questions to consider:
1. What is the geometric interpretation of coincident lines?
2. How can we find the slope of a line from its equation?
3. What is the general form of a line passing through two points?
4. How do we find the point of intersection for two non-coincident lines?
5. Can we use the concept of parametric equations to describe all lines passing through a point?

**Tip:** When two linear equations represent the same line, they have infinitely many common solutions.

Discover how to find the equations of lines passing through the solution of the linear equations 2x + 3y = 6 and 4x + 6y = 12. Understand the concept of coincident lines and learn why infinitely many lines can pass through any point on the line. Suitable for students in Grades 9-11.

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