https://file.aiquickdraw.com/mathbot/file/1726848527194wgtd7yot.jpg

Dan wrote two systems of equations A and B and concluded that they have the same solution.

Let's analyze the two systems of equations given in the image to check Dan's conclusion:

### System A:
\[
\begin{cases}
4x + 3y = 17 \\
2x + 5y = 19
\end{cases}
\]

### System B:
\[
\begin{cases}
8x + 6y = 34 \\
6x + 8y = 36
\end{cases}
\]

Now, let's check if both systems have the same solution.

### 1. System A (Solving by substitution or elimination):
From the first equation:
\[
4x + 3y = 17
\]
From the second equation:
\[
2x + 5y = 19
\]

We'll solve these using elimination or substitution methods to find the solution for \(x\) and \(y\).

### 2. System B (Checking dependency):
The second system appears to have equations that are multiples of each other. We can divide both equations in system B by constants to simplify:

- Divide the first equation \(8x + 6y = 34\) by 2:
\[
4x + 3y = 17
\]

- Divide the second equation \(6x + 8y = 36\) by 2:
\[
3x + 4y = 18
\]

This simplified system is not identical to System A, indicating that System B may not have the same solution. Let's proceed by solving both systems to verify. I'll calculate this now.Both systems, A and B, have the same solution:

\[
x = 2, \quad y = 3
\]

Dan's conclusion is correct. Despite the second system appearing to be different, it simplifies in a way that results in the same solution.

Would you like more details on the solution process or any specific part of the calculation?

Here are some related questions for further exploration:
1. How can we determine if two systems are equivalent before solving?
2. What other methods can we use to solve systems of equations?
3. Why does System B simplify to the same solution even though it looks different initially?
4. How do you check for consistency in a system of linear equations?
5. Can systems with more than two variables have unique solutions like this?

**Tip**: Always check if a system of equations is a multiple of another system, as this can help identify equivalent systems quickly.

Analyze whether two systems of linear equations, A and B, have the same solution. Learn the elimination and substitution methods to solve systems of equations, and explore consistency and dependency in linear systems. Suitable for high school students (Grades 9-12).

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