Question

solve the system by the substitution method.\

$$\begin{cases}y = -3x - 8\\\\3x - 4y = 17\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\bigcirc\\) a. there is exactly one solution. the solution set is \\(\\{\\ \\}\\).\\(\quad\\) (simplify your answer. type an ordered pair.)\\(\bigcirc\\) b. there are infinitely many solutions. the solution set is \\(\\{(x,y)|y = -3x - 8\\}\\) or \\(\\{(x,y)|3x - 4y = 17\\}.\\)\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Substitute \( y = -3x - 8 \) into \( 3x - 4y = 17 \)

Substitute \( y \) in the second equation: \( 3x - 4(-3x - 8) = 17 \)

Step2: Simplify the equation

Expand the left side: \( 3x + 12x + 32 = 17 \)
Combine like terms: \( 15x + 32 = 17 \)
Subtract 32 from both sides: \( 15x = 17 - 32 = -15 \)
Divide by 15: \( x = \frac{-15}{15} = -1 \)

Step3: Find \( y \) using \( y = -3x - 8 \)

Substitute \( x = -1 \): \( y = -3(-1) - 8 = 3 - 8 = -5 \)

Answer:

A. There is exactly one solution. The solution set is \((-1, -5)\)