Identify Linear Functions Among Given Equations: Which Are Parallel?

Which of the following represent linear functions and parallel lines?

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