Question

① $y = -x - 3$ $y = x - 1$ answer: ② $2x + 2y = -6$ $y = -x - 3$ answer: ③ $y = -x - 3$ $2x + 2y = 4$ answer: ④ $2x + 2y = -6$ $3x + y = 3$ answer:

Explanation:

Problem 1: Solve the system \(

$$\begin{cases} y = -x - 3 \\ y = x - 1 \end{cases}$$

\)

Step 1: Set the equations equal

Since both equal \( y \), set \( -x - 3 = x - 1 \).

Step 2: Solve for \( x \)

Add \( x \) to both sides: \( -3 = 2x - 1 \).
Add 1 to both sides: \( -2 = 2x \).
Divide by 2: \( x = -1 \).

Step 3: Find \( y \)

Substitute \( x = -1 \) into \( y = x - 1 \): \( y = -1 - 1 = -2 \).

Step 1: Substitute \( y \) into the first equation

Substitute \( y = -x - 3 \) into \( 2x + 2y = -6 \):
\( 2x + 2(-x - 3) = -6 \).

Step 2: Simplify

Expand: \( 2x - 2x - 6 = -6 \).
Simplify: \( -6 = -6 \).
This is an identity, so infinitely many solutions (the lines are coincident).

Step 1: Substitute \( y \) into the second equation

Substitute \( y = -x - 3 \) into \( 2x + 2y = 4 \):
\( 2x + 2(-x - 3) = 4 \).

Step 2: Simplify

Expand: \( 2x - 2x - 6 = 4 \).
Simplify: \( -6 = 4 \).
This is a contradiction, so no solution (parallel lines).

Answer:

\( x = -1, y = -2 \)

Problem 2: Solve the system \(

$$\begin{cases} 2x + 2y = -6 \\ y = -x - 3 \end{cases}$$

\)