The line mx + ny = 1 passes through the points (1, 2) and (2, 1). What is the value of m?

Options

A. 1
B. 3
C. `(1)/(2)`
D. `(1)/(3)`

Answer

To find the value of \( m \) in the equation \( mx + ny = 1 \) that passes through the points \( (1, 2) \) and \( (2, 1) \), we can follow these steps:

### Step 1: Substitute the first point into the equation
Since the line passes through the point \( (1, 2) \), we can substitute \( x = 1 \) and \( y = 2 \) into the equation \( mx + ny = 1 \).

\[ m(1) + n(2) = 1 \]

This simplifies to:

\[ m + 2n = 1 \quad \text{(Equation 1)} \]

### Step 2: Substitute the second point into the equation
Next, we substitute the second point \( (2, 1) \) into the same equation.

\[ m(2) + n(1) = 1 \]

This simplifies to:

\[ 2m + n = 1 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:

1. \( m + 2n = 1 \) (Equation 1)
2. \( 2m + n = 1 \) (Equation 2)

To eliminate \( n \), we can manipulate these equations. We can multiply Equation 2 by 2 to align the coefficients of \( n \):

\[ 4m + 2n = 2 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 1 from Equation 3
Now we subtract Equation 1 from Equation 3:

\[ (4m + 2n) - (m + 2n) = 2 - 1 \]

This simplifies to:

\[ 4m + 2n - m - 2n = 1 \]

Which further simplifies to:

\[ 3m = 1 \]

### Step 5: Solve for \( m \)
Now, divide both sides by 3:

\[ m = \frac{1}{3} \]

### Conclusion
Thus, the value of \( m \) is \( \frac{1}{3} \).