The equation \( y = -3x + 1 \) represents a straight line in a Cartesian grid (or coordinate system). This line can intersect the grid at infinitely many points, as any real number value for \( x \) will produce a corresponding value for \( y \).
If you're considering "solutions" in terms of integer grid points (lattice points with integer coordinates), you can find specific solutions by substituting integer values for \( x \) into the equation and calculating \( y \).
To determine integer solutions, set \( y \) as an integer:
- Rearranging the equation gives \( y - 1 = -3x \).
- This implies \( x = \frac{1 - y}{3} \).
For \( x \) to also be an integer, \( 1 - y \) must be divisible by 3. This gives us a condition on \( y \): it must be congruent to 1 modulo 3.
The solutions will be in the form of:
- \( y = 1 \) (when \( x = 0 \)),
- \( y = 4 \) (when \( x = -1 \)),
- \( y = 7 \) (when \( x = -2 \)),
- \( y = -2 \) (when \( x = 1 \)),
- and so on.
Thus, there are infinitely many integer solutions where \( y \) satisfies the condition \( y \equiv 1 \mod 3 \).
In conclusion, the line \( y = -3x + 1 \) has:
- Infinitely many solutions in the set of real numbers,
- Infinitely many integer solutions under the condition \( y \equiv 1 \mod 3 \).