The value of expression `mx-ny` is 3 when `x=5` and `y=6`. And its value is 8 when `x=6` and `y=5`. Find the values of m and n.

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) given the expressions for \( mx - ny \) at two different points.

### Step 1: Set up the equations
From the problem statement, we have two scenarios:

1. When \( x = 5 \) and \( y = 6 \), the expression \( mx - ny = 3 \):
\[ m(5) - n(6) = 3 \quad \text{(Equation 1)} \]
This simplifies to:
\[ 5m - 6n = 3 \]

2. When \( x = 6 \) and \( y = 5 \), the expression \( mx - ny = 8 \):
\[ m(6) - n(5) = 8 \quad \text{(Equation 2)} \]
This simplifies to:
\[ 6m - 5n = 8 \]

### Step 2: Solve the system of equations
Now we have the following system of equations:
1. \( 5m - 6n = 3 \) (1)
2. \( 6m - 5n = 8 \) (2)

We can solve these equations using substitution or elimination. Here, we will use substitution.

### Step 3: Express \( m \) in terms of \( n \)
From Equation (1):
\[ 5m = 6n + 3 \]
Dividing by 5:
\[ m = \frac{6n + 3}{5} \quad \text{(Equation 3)} \]

### Step 4: Substitute \( m \) in Equation (2)
Now, substitute Equation (3) into Equation (2):
\[ 6\left(\frac{6n + 3}{5}\right) - 5n = 8 \]
Multiply through by 5 to eliminate the fraction:
\[ 6(6n + 3) - 25n = 40 \]
Expanding this gives:
\[ 36n + 18 - 25n = 40 \]
Combine like terms:
\[ 11n + 18 = 40 \]
Subtract 18 from both sides:
\[ 11n = 22 \]
Divide by 11:
\[ n = 2 \]

### Step 5: Find \( m \)
Now substitute \( n = 2 \) back into Equation (3) to find \( m \):
\[ m = \frac{6(2) + 3}{5} = \frac{12 + 3}{5} = \frac{15}{5} = 3 \]

### Final Answer
The values of \( m \) and \( n \) are:
\[ m = 3, \quad n = 2 \]