Question

The system of equations y=4x y-4=4(x-1) has
.
no solution
infinetley many solutions
one solution

Answer 1

To determine whether the system of equations has no solution, infinitely many solutions, or one solution, we will analyze the two equations given:

1. \( y = 4x \)
2. \( y - 4 = 4(x - 1) \)

Let's simplify the second equation:

Starting with:
\[
y - 4 = 4(x - 1)
\]
Expanding the right side:
\[
y - 4 = 4x - 4
\]
Now, adding 4 to both sides gives:
\[
y = 4x
\]

Now we see that both equations are actually the same:

1. \( y = 4x \)
2. \( y = 4x \)

Since both equations are identical, they represent the same line in a two-dimensional space. Therefore, there are infinitely many solutions since every point on the line described by the equation \( y = 4x \) is a solution to the system.

Thus, the answer is **infinitely many solutions**.