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To solve the problem, we need to find the value of \( m \) such that the system of equations has no solution. The given system of equations is:

\[
\begin{aligned}
(m - 2)x + (m - 3)y &= 9 \\
5x + 6y &= 3
\end{aligned}
\]

For a system of two linear equations to have no solution, the lines represented by the equations must be parallel. This happens when the ratios of the coefficients of \( x \) and \( y \) in the two equations are equal, but the constant terms are not. 

The first equation is: 
\[
(m - 2)x + (m - 3)y = 9
\]
The second equation is:
\[
5x + 6y = 3
\]

To find when the system has no solution, we set up the condition for parallelism:
\[
\frac{m - 2}{5} = \frac{m - 3}{6}
\]

Let's solve this equation for \( m \).

\[
6(m - 2) = 5(m - 3)
\]
Expanding both sides:
\[
6m - 12 = 5m - 15
\]
Solving for \( m \):
\[
6m - 5m = -15 + 12
\]
\[
m = -3
\]

Thus, the value of \( m \) is \( -3 \).

Would you like further details, or have any questions?

Here are 5 related questions:
1. What happens geometrically when two lines are parallel?
2. Can you find the value of \( m \) if the system has an infinite number of solutions?
3. How do you determine if a system of equations is consistent or inconsistent?
4. What are the conditions for two lines to intersect at a single point?
5. How do you solve a system of equations using the substitution method?

**Tip**: When solving a system of linear equations, always check if the system is consistent (one solution), inconsistent (no solution), or dependent (infinite solutions).

Halla m si el sistema no tiene solución. (m-2)x + (m-3)y = 9, 5x + 6y = 3

Learn how to find the value of m such that the system of linear equations (m-2)x + (m-3)y = 9 and 5x + 6y = 3 has no solution. This problem covers key algebraic concepts, including systems of equations and the condition for parallel lines. Suitable for students in Grades 9-11.

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