We have the system of linear equations:

1. \( 2x + 3y = 17 \)
2. \( 3x - 2y = 6 \)

To solve this, we can use either the substitution method or the elimination method. Let's use the elimination method to eliminate one of the variables.

### Step 1: Multiply the equations to align coefficients of one variable
We will aim to eliminate \( y \) by multiplying the first equation by 2 and the second equation by 3, so that the coefficients of \( y \) become the same (but opposite in sign).

- Multiply the first equation by 2:
  \[
  2(2x + 3y) = 2(17) \implies 4x + 6y = 34
  \]
- Multiply the second equation by 3:
  \[
  3(3x - 2y) = 3(6) \implies 9x - 6y = 18
  \]

Now we have:
\[
4x + 6y = 34 \quad \text{(equation 3)}
\]
\[
9x - 6y = 18 \quad \text{(equation 4)}
\]

### Step 2: Add the equations to eliminate \( y \)
By adding equations 3 and 4:
\[
(4x + 6y) + (9x - 6y) = 34 + 18
\]
This simplifies to:
\[
13x = 52
\]
Solving for \( x \):
\[
x = \frac{52}{13} = 4
\]

### Step 3: Substitute \( x = 4 \) into one of the original equations
Substitute \( x = 4 \) into the first equation \( 2x + 3y = 17 \):
\[
2(4) + 3y = 17
\]
This simplifies to:
\[
8 + 3y = 17
\]
Solving for \( y \):
\[
3y = 17 - 8 = 9 \quad \Rightarrow \quad y = \frac{9}{3} = 3
\]

### Final Solution:
The solution to the system of equations is:
\[
x = 4, \quad y = 3
\]

Would you like further details or clarification on any step?

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Here are 5 related questions to expand on this concept:

1. How would you solve the same system of equations using substitution?
2. Can you solve another system, like \( x + y = 5 \) and \( 2x - 3y = 4 \)?
3. How can you verify that the solution \( x = 4 \) and \( y = 3 \) is correct?
4. What happens if both equations have the same slope? Will they have a solution?
5. How would the solution process change if there were three equations and three variables?

**Tip:** When using elimination, aligning the coefficients is crucial for effectively canceling one variable.

Learn how to solve the system of linear equations 2x + 3y = 17 and 3x - 2y = 6 using the elimination method. This step-by-step solution covers key algebra concepts and techniques for aligning coefficients and solving for variables. Suitable for students in Grades 8-10.

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