write an equation of a line passing through the point representing solution of the pair of linear equations 3x-5y=20 and 6x-10y=-40 how many such lines can be find

To solve this, we need to first determine the point where the two given linear equations intersect. Once we have this point, we can write the equation of a line passing through it.

The given system of equations is:

1. \( 3x - 5y = 20 \)
2. \( 6x - 10y = -40 \)

### Step 1: Solve the system of equations

We observe that the second equation \( 6x - 10y = -40 \) is just a multiple of the first equation. Divide the second equation by 2:

\[
\frac{6x - 10y}{2} = \frac{-40}{2}
\]
\[
3x - 5y = -20
\]

So, we now have the system:
\[
3x - 5y = 20 \quad \text{(Equation 1)}
\]
\[
3x - 5y = -20 \quad \text{(Equation 2)}
\]

This system is inconsistent because the same term \( 3x - 5y \) cannot be equal to both 20 and -20 at the same time. Thus, **there is no solution to this system of equations**. The two lines represented by these equations are parallel and will never intersect.

### Step 2: How many lines can be found passing through the intersection point?

Since the two lines do not intersect, there is no common solution point. Therefore, **there are no lines that can be found passing through a common point** because the two lines are parallel and never meet.

### Conclusion:

- **There is no point of intersection** of the two lines.
- Therefore, **no lines can be found** passing through a solution point, because the system is inconsistent.

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Would you like further details on parallel lines or related concepts? 

Here are 5 questions that may expand on this topic:
1. How do we identify parallel lines from their equations?
2. What are inconsistent systems of linear equations?
3. What does it mean when two lines are coincident instead of parallel?
4. Can two lines with different slopes ever be parallel?
5. How do we find the distance between two parallel lines?

**Tip**: When two equations are multiples of each other but their constants are different, they represent parallel lines.

Understand why the system of linear equations 3x - 5y = 20 and 6x - 10y = -40 has no solution. Learn about inconsistent systems and parallel lines with a step-by-step solution. Suitable for Grades 9-11 students learning about systems of equations and their solutions.

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