https://file.aiquickdraw.com/mathbot/file/1727874734505rlghez6y.jpg

The system of linear equations in the image is:

\[
3x + 2y = 14
\]
\[
2x + 3y = 14
\]

We can solve this system using either the substitution method, the elimination method, or matrix methods. Let's proceed with the elimination method:

1. Multiply the first equation by 3 and the second by 2 to align the coefficients of \( y \):

\[
(3x + 2y = 14) \times 3 \implies 9x + 6y = 42
\]
\[
(2x + 3y = 14) \times 2 \implies 4x + 6y = 28
\]

2. Subtract the second equation from the first to eliminate \( y \):

\[
(9x + 6y) - (4x + 6y) = 42 - 28
\]
\[
5x = 14
\]

3. Solve for \( x \):

\[
x = \frac{14}{5} = 2.8
\]

4. Substitute \( x = 2.8 \) into one of the original equations to find \( y \). Using the first equation:

\[
3(2.8) + 2y = 14
\]
\[
8.4 + 2y = 14
\]
\[
2y = 14 - 8.4 = 5.6
\]
\[
y = \frac{5.6}{2} = 2.8
\]

Thus, the solution is:

\[
x = 2.8, \quad y = 2.8
\]

Would you like any further details or have any questions about this method?

Here are some related questions for further exploration:
1. What are the advantages of using the elimination method over substitution?
2. How would you solve the system using matrices (Cramer's rule or Gaussian elimination)?
3. What happens if the system has no solution or an infinite number of solutions?
4. How can graphing the system help verify the solution?
5. Can you extend this method to solve systems with three variables?

**Tip**: Always check your solution by substituting the values of \( x \) and \( y \) back into both original equations!

Solve the system of linear equations: 3x + 2y = 14 and 2x + 3y = 14.

This page provides a detailed solution to the system of linear equations 3x + 2y = 14 and 2x + 3y = 14 using the elimination method. Suitable for students in Grades 8-10, this problem helps develop key algebraic skills.

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