The number of integral values of `m` for which the x-coordinate of the point of intersection of the lines `3x+4y=9` and `y=m x+1` is also an integer is 2 (b) 0 (c) 4 (d) 1

Options

A. 2
B. 0
C. 4
D. 1

Explanation

To solve the problem, we need to find the number of integral values of \( m \) for which the x-coordinate of the intersection of the lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is an integer.

### Step 1: Set up the equations
We have two equations:
1. \( 3x + 4y = 9 \)
2. \( y = mx + 1 \)

### Step 2: Substitute \( y \) from the second equation into the first equation
Substituting \( y = mx + 1 \) into \( 3x + 4y = 9 \):
\[ 3x + 4(mx + 1) = 9 \]
This simplifies to:
\[ 3x + 4mx + 4 = 9 \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ 3x + 4mx = 9 - 4 \]
\[ 3x + 4mx = 5 \]

### Step 4: Factor out \( x \)
Factoring out \( x \):
\[ x(3 + 4m) = 5 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{5}{3 + 4m} \]

### Step 6: Determine when \( x \) is an integer
For \( x \) to be an integer, \( 3 + 4m \) must be a divisor of 5. The divisors of 5 are \( \pm 1 \) and \( \pm 5 \).

### Step 7: Set up equations for each divisor
We can set up equations based on the divisors:
1. \( 3 + 4m = 1 \)
2. \( 3 + 4m = -1 \)
3. \( 3 + 4m = 5 \)
4. \( 3 + 4m = -5 \)

### Step 8: Solve each equation for \( m \)

1. **For \( 3 + 4m = 1 \)**:
\[ 4m = 1 - 3 \implies 4m = -2 \implies m = -\frac{1}{2} \]

2. **For \( 3 + 4m = -1 \)**:
\[ 4m = -1 - 3 \implies 4m = -4 \implies m = -1 \]

3. **For \( 3 + 4m = 5 \)**:
\[ 4m = 5 - 3 \implies 4m = 2 \implies m = \frac{1}{2} \]

4. **For \( 3 + 4m = -5 \)**:
\[ 4m = -5 - 3 \implies 4m = -8 \implies m = -2 \]

### Step 9: Identify integral values of \( m \)
From the calculations, the integral values of \( m \) are:
- \( m = -1 \)
- \( m = -2 \)

### Step 10: Count the integral values
Thus, the number of integral values of \( m \) is **2**.

### Final Answer
The number of integral values of \( m \) for which the x-coordinate of the point of intersection is an integer is **2**.

---