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`

Explanation

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

### Step 1: Substitute the first point into the equation
We start by substituting the first point \( (1, 2) \) 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)

We can solve these equations simultaneously. Let's express \( n \) from Equation 1:

\[ 2n = 1 - m \implies n = \frac{1 - m}{2} \quad \text{(Substituting into Equation 2)} \]

Now, substitute \( n \) into Equation 2:

\[ 2m + \left(\frac{1 - m}{2}\right) = 1 \]

### Step 4: Clear the fraction
To eliminate the fraction, multiply the entire equation by 2:

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

This simplifies to:

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

### Step 5: Combine like terms
Combine the terms involving \( m \):

\[ 3m + 1 = 2 \]

### Step 6: Isolate \( m \)
Now, isolate \( m \):

\[ 3m = 2 - 1 \]
\[ 3m = 1 \]
\[ m = \frac{1}{3} \]

### Final Answer
Thus, the value of \( m \) is:

\[ \boxed{\frac{1}{3}} \]

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