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

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

Answer: (A)

Verified step by step guidance

1. Start by writing down the system of equations clearly: \[\begin{cases} y + 2x = 8 \\ y - 5x = -6 \end{cases}\]
2. To determine the number of solutions without graphing, use the substitution or elimination method. Here, subtract the second equation from the first to eliminate y: \[ (y + 2x) - (y - 5x) = 8 - (-6) \]
3. Simplify the left side by canceling y and combining like terms: \[ y + 2x - y + 5x = 8 + 6 \] which simplifies to \[ 7x = 14 \]
4. Solve for x by dividing both sides by 7: \[ x = \frac{14}{7} \]
5. Substitute the value of x back into either original equation to find y, confirming that the system has exactly one solution since both equations intersect at a single point.