Question

solve the system by the method of your choice.
\

$$\begin{cases} y = -4x - 3 \\\\ -12x - 9 = 3y \\end{cases}$$

Explanation:

Step1: Substitute \( y = -4x - 3 \) into the second equation

We have the second equation \( -12x - 9 = 3y \). Substitute \( y \) with \( -4x - 3 \):
\( -12x - 9 = 3(-4x - 3) \)

Step2: Simplify the right - hand side

Expand the right - hand side: \( 3(-4x - 3)=-12x - 9 \)
So the equation becomes \( -12x - 9=-12x - 9 \)

Step3: Analyze the resulting equation

If we add \( 12x+9 \) to both sides of the equation \( -12x - 9=-12x - 9 \), we get \( 0 = 0 \)

This means that the two equations are dependent (they represent the same line), and there are infinitely many solutions. The solutions are all the points \((x,y)\) such that \( y=-4x - 3 \) (or equivalently, \( -12x - 9 = 3y \)).

Answer:

The system has infinitely many solutions, and the solutions are given by the equation \( y=-4x - 3 \) (or \( -12x - 9 = 3y \)) where \( x \) is any real number.