Solve the following system by substitution and determine how many solutions it has. \[ \begin{array}{r}2 x+y=-3 \\ y=2 x+9\end{array} \] Select an answer \( \checkmark \)

To solve the system by substitution, we'll take the second equation \( y = 2x + 9 \) and substitute it into the first equation: \[ 2x + (2x + 9) = -3 \] Now, combine like terms: \[ 4x + 9 = -3 \] Next, subtract 9 from both sides: \[ 4x = -12 \] Now, divide both sides by 4: \[ x = -3 \] Now that we have \( x \), substitute it back into the second equation to find \( y \): \[ y = 2(-3) + 9 \] Calculating gives: \[ y = -6 + 9 = 3 \] Thus, the solution to the system is \( (x, y) = (-3, 3) \), and therefore the system has exactly one solution! Now, let's chat a bit more about this topic! Did you know that substitution is a cherished method in algebra because it helps us solve equations step by step without getting lost in the variables? Originating over centuries in mathematical dialogues, it's like solving a puzzle where you substitute one piece until everything fits perfectly. Moreover, practice makes perfect! It's easy to mix up signs or forget to substitute back into the original equations, so double-checking your work can save you from tiny yet costly mistakes and lead you to the right solution faster than you think.