Question

solve each system of linear equations.

1. \

$$\begin{cases} y + 2x + 4 = 0 \\\\ 2x = -y - 4 \\end{cases}$$

2.

Explanation:

Step1: Simplify the first equation

We start with the first equation \( y + 2x + 4 = 0 \). Let's rearrange it to solve for \( y \). Subtract \( 2x \) and \( 4 \) from both sides:
\( y = -2x - 4 \)

Step2: Analyze the second equation

Now look at the second equation \( 2x = -y - 4 \). Let's rearrange this to solve for \( y \) as well. Add \( y \) to both sides and add \( 4 \) to both sides:
\( y + 2x = -4 \)
Then subtract \( 2x \) from both sides:
\( y = -2x - 4 \)

Step3: Determine the relationship between the equations

We can see that both equations simplify to \( y = -2x - 4 \). This means that the two lines are coincident (they are the same line).

Step4: Describe the solution set

For a system of linear equations where the two equations represent the same line, there are infinitely many solutions. Any point \( (x, y) \) that satisfies \( y = -2x - 4 \) is a solution. We can express the solution as \( y = -2x - 4 \) where \( x \) is any real number, or in set notation: \( \{(x, -2x - 4) | x \in \mathbb{R}\} \)

Answer:

The system has infinitely many solutions, and the solution can be expressed as \( y = -2x - 4 \) (where \( x \) is any real number) or in set notation \( \{(x, -2x - 4) | x \in \mathbb{R}\} \)