how many solutions does y=-3x+1 have in a grid

Answer 1

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:

1. Rearranging the equation gives $y-1=-3x$.
2. 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 $y\equiv 1\mathrm{mod}3$.

In conclusion, the line $y=-3x+1$ has:

• Infinitely many solutions in the set of real numbers,
• Infinitely many integer solutions under the condition $y\equiv 1\mathrm{mod}3$.