The number of intergral values of k if x coordinate of the point of intersection of the line `5x+7y=12and y=kx+2` is also an integer is

Answer

To find the number of integral values of \( k \) such that the x-coordinate of the point of intersection of the lines \( 5x + 7y = 12 \) and \( y = kx + 2 \) is an integer, we can follow these steps:

### Step 1: Rewrite the equations
We have two equations:
1. \( 5x + 7y = 12 \)
2. \( y = kx + 2 \)

We can rewrite the second equation in standard form:
\[ kx - y + 2 = 0 \quad \text{or} \quad kx - y = -2 \]

### Step 2: Find the point of intersection
To find the point of intersection, we can solve these two equations simultaneously. We can substitute \( y \) from the second equation into the first equation:
\[ 5x + 7(kx + 2) = 12 \]
Expanding this gives:
\[ 5x + 7kx + 14 = 12 \]
Combining like terms:
\[ (5 + 7k)x + 14 = 12 \]
Now, isolate \( x \):
\[ (5 + 7k)x = 12 - 14 \]
\[ (5 + 7k)x = -2 \]
Thus, we can express \( x \) as:
\[ x = \frac{-2}{5 + 7k} \]

### Step 3: Determine when \( x \) is an integer
For \( x \) to be an integer, the denominator \( 5 + 7k \) must divide \(-2\). The integer divisors of \(-2\) are \(-2, -1, 1, 2\).

### Step 4: Set up equations for each divisor
We will set \( 5 + 7k \) equal to each divisor and solve for \( k \):

1. **Case 1:** \( 5 + 7k = -2 \)
\[ 7k = -2 - 5 \implies 7k = -7 \implies k = -1 \]

2. **Case 2:** \( 5 + 7k = -1 \)
\[ 7k = -1 - 5 \implies 7k = -6 \implies k = -\frac{6}{7} \]

3. **Case 3:** \( 5 + 7k = 1 \)
\[ 7k = 1 - 5 \implies 7k = -4 \implies k = -\frac{4}{7} \]

4. **Case 4:** \( 5 + 7k = 2 \)
\[ 7k = 2 - 5 \implies 7k = -3 \implies k = -\frac{3}{7} \]

### Step 5: Identify integral values of \( k \)
From the cases above, the only integral value of \( k \) is:
- \( k = -1 \)

The other values \( -\frac{6}{7}, -\frac{4}{7}, -\frac{3}{7} \) are not integers.

### Conclusion
Thus, the number of integral values of \( k \) such that the x-coordinate of the intersection point is an integer is:
\[ \boxed{1} \]