Without using a graph, determine the number of solutions in the following systems of equations, then classify the system.

• A. Independent and Consistent
• B. Dependent and Consistent
• C. Inconsistent

Answer: (A)

Verified step by step guidance

1. Step 1: Write down the system of equations clearly: \[\begin{cases} y + 2x = 8 \\ y - 5x = -6 \end{cases}\]
2. Step 2: Use substitution or elimination to analyze the system. Here, both equations are solved for y, so set the right-hand sides equal to each other: \[y + 2x = 8 \implies y = 8 - 2x\] \[y - 5x = -6 \implies y = -6 + 5x\] Set equal: \[8 - 2x = -6 + 5x\]
3. Step 3: Solve the equation for x: \[8 - 2x = -6 + 5x\] Add 2x to both sides and add 6 to both sides: \[8 + 6 = 5x + 2x\] \[14 = 7x\] Divide both sides by 7: \[x = 2\]
4. Step 4: Substitute the value of x back into one of the original equations to find y: Using \[y = 8 - 2x\]: \[y = 8 - 2(2) = 8 - 4\] \[y = 4\]
5. Step 5: Since there is exactly one solution (x=2, y=4), the system is consistent (has at least one solution) and independent (the equations represent two distinct lines).