Identify Linear Functions Among Given Equations: Which Are Parallel?

To solve this problem, we'll analyze each pair of given equations to see if they are linear and parallel.

Let's examine each pair:

• Choice 1:
  $y = \frac{1}{2}x + 10$
  $y = \frac{1}{2}(x + 2)$ simplifies to $y = \frac{1}{2}x + 1$
  Both equations are linear with the same slope of $\frac{1}{2}$, indicating they are parallel.
• Choice 2:
  $y = 3(x + 4)$ simplifies to $y = 3x + 12$
  $y = 3x^2 + 12$ is not in the form $y = mx + b$ as it includes an $x^2$ term. Thus, it is non-linear.
• Choice 3:
  $y = 5 + 12x$ is already in the form $y = mx + b$ with $m = 12$
  $y = 5 + 12 + x$ simplifies to $y = x + 17$, which has a slope of 1.
  Slopes are different, so not parallel.
• Choice 4:
  $y = 3x + 2$, slope $m = 3$
  $y = 2x + 3$, slope $m = 2$
  Different slopes, thus not parallel.

Therefore, based on our analysis, the correct choice is Choice 1:

$y = \frac{1}{2}x + 10$ and $y = \frac{1}{2}(x + 2)$