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 = mx + 1` is also an integer, is

Options

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

Answer

To find the number of integral values of \( m \) for which the x-coordinate of the point of intersection of the lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is an integer, we can follow these steps:

### Step 1: Find the point of intersection of the two lines.
We have the equations:
1. \( 3x + 4y = 9 \)
2. \( y = mx + 1 \)

Substituting the second equation into the first:

\[ 3x + 4(mx + 1) = 9 \]

### Step 2: Simplify the equation.
Expanding the equation:

\[ 3x + 4mx + 4 = 9 \]

Combine like terms:

\[ (3 + 4m)x + 4 = 9 \]

### Step 3: Rearranging the equation.
Now, isolate \( x \):

\[ (3 + 4m)x = 9 - 4 \]
\[ (3 + 4m)x = 5 \]
\[ x = \frac{5}{3 + 4m} \]

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

### Step 5: Set up equations based on the divisors.
1. \( 3 + 4m = 1 \)
\[ 4m = 1 - 3 \implies 4m = -2 \implies m = -\frac{1}{2} \quad (\text{not an integer}) \]

2. \( 3 + 4m = -1 \)
\[ 4m = -1 - 3 \implies 4m = -4 \implies m = -1 \quad (\text{integer}) \]

3. \( 3 + 4m = 5 \)
\[ 4m = 5 - 3 \implies 4m = 2 \implies m = \frac{1}{2} \quad (\text{not an integer}) \]

4. \( 3 + 4m = -5 \)
\[ 4m = -5 - 3 \implies 4m = -8 \implies m = -2 \quad (\text{integer}) \]

### Step 6: List the integral values of \( m \).
The integral values of \( m \) we found are:
- \( m = -1 \)
- \( m = -2 \)

### Step 7: Count the integral values.
Thus, the number of integral values of \( m \) is \( 2 \).

### Final Answer:
The number of integral values of \( m \) is \( \boxed{2} \).

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