Identifying Parallel Lines: Understanding Linear Function Slopes

To solve this problem, we'll examine each pair of equations to determine which consists of linear functions with parallel lines.

• Step 1: Identify the form of the equation for each choice and ensure they are linear if they can be written as $y = mx + b$.

• Step 2: Calculate or identify the slope for each equation to compare within the pair.

Now, consider each given choice:

Choice 1:
- Equation 1: $y = 5x$ has a slope of 5.
- Equation 2: $y = 5(x^2 + 1)$ simplifies to a nonlinear form because of the $x^2$ term, so it is not relevant for parallelism in linear functions.

Choice 2:
- Equation 1: $y = x$ has a slope of 1. - Equation 2: $y = -x + x + 10$ simplifies to $y = 10$ which is a constant and does not form a linear equation with variable terms, thus irrelevant.

Choice 3:
- Equation 1: $y = 2(x + 3)$ simplifies to $y = 2x + 6$, slope is 2.
- Equation 2: $y = -2(3 + x)$, simplifies to $y = -6 - 2x$, slope is -2.
- Slopes are not equal, lines are not parallel.

Choice 4:
- Equation 1: $y = -4(x + 1)$ simplifies to $y = -4x - 4$, slope is -4.
- Equation 2: $y = 8x - 12(x + 1)$ simplifies to:
$y = 8x - 12x - 12$, which simplifies to $y = -4x - 12$, slope is -4.
Both slopes are -4, indicating these are parallel lines.

Therefore, the correct choice is Choice 4:

_^$y=-4(x+1)$

_{^{$y=8x-12(x+1)$}}