Question: Solve the system by substitution. If the system is inconsistent or has dependent equations, say so…

Solve the system by substitution. If the system is inconsistent or has dependent equations, say so.

\[ -x - 2y = -7 \\ y = 8x - 22 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The system has a single solution. The solution set is \((\boxed{ }, \boxed{ })\). (Type an ordered pair. Type integers or simplified fractions.)

B. There are infinitely many solutions and the equations are dependent. The solution set is \(\{(x, y) | -x - 2y = -7\}\)

C. The system is inconsistent. The solution set is the empty set.

Solution

Step 1 From the second equation, express \( y \) in terms of \( x \): \[ y = 8x - 22 \] Step 2 Substitute \( y = 8x - 22 \) into the first equation: \[ - x - 2(8x - 22) = -7 & \] Step 3 Distribute and simplify: \[ - x - 16x + 44 = -7 & \] Step 4 Combine like terms: \[ -17x + 44 = -7 & \] Step 5 Subtract 44 from both sides: \[ -17x = -51 & \] Step 6 Divide both sides by -17: \[ x = 3 & \] Step 7 Substitute \( x = 3 \) back into \( y = 8x - 22 \): \[ y = 8(3) - 22 & \] Step 8 Calculate \( y \): \[ y = 24 - 22 = 2 & \] The solution set is \( (3, 2) \). Answer: A. The system has a single solution. The solution set is \( (3, 2) \).