Question

solve the given system by the method of your choice.\

$$\begin{cases}x = y + 12\\\\4x + 5y = -6\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\a. there is exactly one solution. the solution set is \\{6, -6\\}.\\(\text{(simplify your answer. type an ordered pair.)}\\)\\b. there are infinitely many solutions. the solution set is \\{\\(x,y\\)|x = y + 12\\} or \\{\\(x,y\\)|4x + 5y = -6\\}.\\c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Substitute \( x = y + 12 \) into \( 4x + 5y = -6 \)

Substitute \( x \) in the second equation: \( 4(y + 12) + 5y = -6 \)

Step2: Expand and simplify the equation

Expand: \( 4y + 48 + 5y = -6 \)
Combine like terms: \( 9y + 48 = -6 \)
Subtract 48 from both sides: \( 9y = -6 - 48 = -54 \)
Divide by 9: \( y = \frac{-54}{9} = -6 \)

Step3: Find \( x \) using \( x = y + 12 \)

Substitute \( y = -6 \): \( x = -6 + 12 = 6 \)

Answer:

A. There is exactly one solution. The solution set is \(\{6, -6\}\) (where the ordered pair is \((6, -6)\))