How many integer values of x and y satisfy the expression 4x + 7y = 3 where |x|<1000 and |y| < 1000?.

Answer

To solve the equation \(4x + 7y = 3\) for integer values of \(x\) and \(y\) under the constraints \(|x| < 1000\) and \(|y| < 1000\), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 4x + 7y = 3 \]
We can express \(y\) in terms of \(x\):
\[ 7y = 3 - 4x \quad \Rightarrow \quad y = \frac{3 - 4x}{7} \]

### Step 2: Finding Integer Solutions
For \(y\) to be an integer, the numerator \(3 - 4x\) must be divisible by \(7\). This means:
\[ 3 - 4x \equiv 0 \mod{7} \]
Rearranging gives:
\[ 4x \equiv 3 \mod{7} \]

### Step 3: Solving the Modular Equation
To solve \(4x \equiv 3 \mod{7}\), we can find the multiplicative inverse of \(4\) modulo \(7\). The inverse is \(2\) because:
\[ 4 \cdot 2 \equiv 1 \mod{7} \]
Multiplying both sides of the equation \(4x \equiv 3\) by \(2\):
\[ x \equiv 6 \mod{7} \]
This means:
\[ x = 6 + 7k \quad \text{for some integer } k \]

### Step 4: Substituting Back to Find \(y\)
Substituting \(x = 6 + 7k\) back into the equation for \(y\):
\[ y = \frac{3 - 4(6 + 7k)}{7} = \frac{3 - 24 - 28k}{7} = \frac{-21 - 28k}{7} = -3 - 4k \]
Thus, we have:
\[ x = 6 + 7k \quad \text{and} \quad y = -3 - 4k \]

### Step 5: Applying the Constraints
Now we need to apply the constraints \(|x| < 1000\) and \(|y| < 1000\).

1. For \(|x| < 1000\):
\[ -1000 < 6 + 7k < 1000 \]
This simplifies to:
\[ -1006 < 7k < 994 \quad \Rightarrow \quad -144.857 < k < 142 \]
Thus, \(k\) can take integer values from \(-144\) to \(142\).

2. For \(|y| < 1000\):
\[ -1000 < -3 - 4k < 1000 \]
This simplifies to:
\[ -997 < -4k < 1003 \quad \Rightarrow \quad -250.75 < k < -250.75 \]
Thus, \(k\) can take integer values from \(-250\) to \(249\).

### Step 6: Finding the Range of \(k\)
The valid integer values of \(k\) must satisfy both constraints. The overlapping range from \(-144\) to \(142\) gives:
- Minimum \(k = -143\)
- Maximum \(k = 142\)

### Step 7: Counting the Integer Values
The total number of integer values for \(k\) is:
\[ 142 - (-143) + 1 = 142 + 143 + 1 = 286 \]

### Final Answer
Thus, the total number of integer solutions \((x, y)\) that satisfy the equation under the given constraints is:
\[ \boxed{286} \]