https://file.aiquickdraw.com/mathbot/file/1728134516257tza8smmh.jpg

Let's solve the system of linear equations as shown in the image:

1) \( 3x - 4y = 1 \)
2) \( 6x - 8y = -5 \)

### Step 1: Recognize Proportionality
The second equation, \( 6x - 8y = -5 \), is simply the first equation \( 3x - 4y = 1 \) multiplied by 2:

\[
2(3x - 4y) = 2(1)
\]
This simplifies to:
\[
6x - 8y = 2
\]

But the second equation is:
\[
6x - 8y = -5
\]

### Step 2: Check for Inconsistency
We now have:
\[
6x - 8y = 2
\]
and
\[
6x - 8y = -5
\]

These two equations contradict each other, which means that there is **no solution** to this system. The system is inconsistent.

### Conclusion:
There is no solution to the given system of equations. The lines are parallel and do not intersect.

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Would you like further details or have any questions?

Here are five related questions you could explore:

1. What does it mean geometrically when two equations in a system have no solution?
2. How can you identify if a system is consistent or inconsistent just by looking at it?
3. What methods can you use to solve a consistent system of linear equations?
4. How would the system change if the second equation were \(6x - 8y = 2\) instead of \(6x - 8y = -5\)?
5. Can you solve this system using substitution or elimination, and how would the method affect the outcome?

**Tip:** Always check if two equations in a system are multiples of each other. If they contradict after simplification, the system is inconsistent.

Find the solution set of the system of linear equations: 1) 3x - 4y = 1, 2) 6x - 8y = -5

Learn how to solve the system of linear equations 3x - 4y = 1 and 6x - 8y = -5. This system is identified as inconsistent due to a proportional relationship between the equations with differing constants, indicating no solution. Suitable for Grades 9-11.

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