Question

Which equation describes an infinite solution system with the equation: y = 4x - 3

Answer 1

An infinite solution system of equations occurs when two equations represent the same line. In order to describe an infinite solution system with the given equation \( y = 4x - 3 \), you can take the original equation and multiply or manipulate it to create a second equation that is equivalent to the first.

For example, you can multiply the entire equation by a non-zero constant. Let's say we multiply by 2:

\[
2y = 2(4x - 3)
\]

This simplifies to:

\[
2y = 8x - 6
\]

Now, if you consider the system of equations:

1. \( y = 4x - 3 \)
2. \( 2y = 8x - 6 \)

These two equations represent the same line and will have infinitely many solutions (all the points on the line). You can also express the second equation in slope-intercept form:

\[
y = 4x - 3
\]

Thus, any multiple of the original equation, or any manipulation that keeps the relationship intact, will yield a second equation resulting in an infinite solution system. For example:

- \( 3y = 12x - 9 \)
- \( -y + 4x = 3 \)

All of these equations represent the same line and, therefore, any system including them will have infinitely many solutions.