Determine the number of solutions the system of equations has without graphing.

• A. $1$
• B. Infinitely Many
• C. $0$

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the slopes and y-intercepts of both equations. The first equation is \(y = \frac{1}{4}x + 11\) and the second is \(y = \frac{1}{4}x - 6\).
2. Step 2: Compare the slopes of the two lines. Both have the same slope, \(\frac{1}{4}\), which means the lines are parallel.
3. Step 3: Compare the y-intercepts. The first line has a y-intercept of 11, and the second has a y-intercept of -6, so they are different.
4. Step 4: Since the lines are parallel (same slope) but have different y-intercepts, they will never intersect.
5. Step 5: Conclude that the system of equations has no solutions because parallel lines do not intersect.