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The answer is no solution but how do you know solve everything I want to all the step leading to no solution

To determine that the system of linear equations has **no solution**, we need to follow through the solving process and observe if the equations lead to a contradiction. Here's the step-by-step approach:

### Given system of equations:
1. \( 3x - 4y = 1 \)
2. \( 6x - 8y = -5 \)

### Step 1: Analyze the system
We can notice that the second equation is a multiple of the first equation, but we will verify it.

The second equation can be written as:
\[
6x - 8y = 2(3x - 4y)
\]
However, the constants on the right-hand side are different. The first equation has a constant of 1, while the second equation's constant is \(-5\).

### Step 2: Eliminate one variable by manipulating the system
We can eliminate one of the variables, say \(x\), using the method of elimination. To do this, we multiply the first equation by 2 to match the coefficients of \(x\) in both equations.

Multiply the first equation by 2:
\[
2(3x - 4y) = 2 \cdot 1
\]
This gives:
\[
6x - 8y = 2
\]

### Step 3: Subtract the two equations
Now, subtract the second equation from this new equation:
\[
(6x - 8y) - (6x - 8y) = 2 - (-5)
\]
On the left-hand side, the terms \(6x - 8y\) cancel out, leaving:
\[
0 = 7
\]

### Step 4: Conclusion
The result \(0 = 7\) is a **contradiction**, meaning there is no value of \(x\) or \(y\) that can satisfy both equations at the same time. This contradiction implies that the system of equations is **inconsistent**, and therefore, there is **no solution**.

### Recap of key steps:
1. Multiply the first equation by 2 to align the coefficients.
2. Subtract the two equations to eliminate one variable.
3. Observe the resulting contradiction \(0 = 7\).

Since the equations lead to an impossible result, we conclude that the system has no solution.

Would you like any more details or explanations on these steps?

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Here are 5 questions to extend your understanding:
1. What are the different methods to solve a system of linear equations?
2. Why does a contradiction in a system of equations indicate no solution?
3. How can you tell visually or graphically that two lines will never intersect?
4. What are consistent, inconsistent, and dependent systems of equations?
5. How would the solution change if the constants were identical?

**Tip:** When solving a system of linear equations, always check if the equations are multiples of each other. If they are, but their constants differ, the system will have no solution.

How do you solve the system of linear equations: 3x - 4y = 1 and 6x - 8y = -5, leading to no solution?

Solve the system of linear equations: 3x - 4y = 1 and 6x - 8y = -5. This step-by-step guide explains why the system is inconsistent and leads to no solution. Learn about linear equations, elimination methods, and contradictions in systems.

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