To determine which function has the same y-intercept as the graph shown, we need to find the y-intercept of the line in the image, then check each equation to see which one shares this y-intercept.

Step-by-Step Solution

1. Identify the y-intercept from the graph: The graph shows a line crossing the y-axis. From visual inspection, it looks like the y-intercept is around $y = -3$.

2. Rewrite the equations in slope-intercept form (y = mx + b) if necessary to identify their y-intercepts:

   • (1) $y = \frac{12 - 6x}{4}$: Simplify this to find $y = 3 - \frac{3x}{2}$, which gives a y-intercept of $3$.
   • (2) $27 + 3y = 6x$: Rearrange to solve for $y$, resulting in $y = 2x - 9$, so the y-intercept is $-9$.
   • (3) $6y + x = 18$: Rearrange to get $y = -\frac{1}{6}x + 3$, giving a y-intercept of $3$.
   • (4) $y + 3 = 6x$: Simplify this to $y = 6x - 3$, which gives a y-intercept of $-3$.

Conclusion

The correct answer is Option (4) $y = 6x - 3$, as it has the same y-intercept of $-3$ as the graph.

Would you like more details on any of these steps?

Here are some follow-up questions that might deepen your understanding:

1. How do you identify the y-intercept of a line from an equation?
2. What are the steps to rearrange an equation to slope-intercept form?
3. Why is it useful to express a line in slope-intercept form?
4. What does the slope of a line represent in an equation?
5. How can you determine the slope and y-intercept directly from a graph?

Tip: Always look for the y-intercept by setting $x = 0$ in an equation when identifying it in non-standard forms.