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.\\(\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)|x = y + 12\\}\\) or \\(\\{(x,y)|4x + 5y = -6\\}\\).\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Substitute \( x = y + 12 \) into the second equation

We have the second equation \( 4x + 5y = -6 \). Replace \( x \) with \( y + 12 \):
\( 4(y + 12) + 5y = -6 \)

Step2: Simplify and solve for \( y \)

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

Step3: Solve for \( x \)

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

Answer:

A. \((6,-6)\)