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 with the given system of equations: \[\begin{cases} y + 2x = 8 \\ y - 5x = -6 \end{cases}\]
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 and right sides: \[ y + 2x - y + 5x = 8 + 6 \] which reduces to \[ 7x = 14 \]
4. Solve for x: \[ x = \frac{14}{7} \]
5. Substitute the value of x back into one of the original equations (for example, the first one) to find y: \[ y + 2\left(\frac{14}{7}\right) = 8 \] Then solve for y to find the corresponding y-value.