Question

a system of equations is given.
y = -3x - 5
y = 3x + 1
part a.
which graph represents the solution of the system?

Explanation:

To determine the graph of the system \(

$$\begin{cases} y = -3x - 5 \\ y = 3x + 1 \end{cases}$$

\), we first find the solution (intersection point) by solving the system.

Step 1: Set the equations equal

Since both equal \( y \), set \( -3x - 5 = 3x + 1 \).

Step 2: Solve for \( x \)

Add \( 3x \) to both sides:
\( -5 = 6x + 1 \)
Subtract \( 1 \) from both sides:
\( -6 = 6x \)
Divide by \( 6 \):
\( x = -1 \)

Step 3: Find \( y \)

Substitute \( x = -1 \) into \( y = 3x + 1 \):
\( y = 3(-1) + 1 = -3 + 1 = -2 \)

The solution is \( (-1, -2) \). To identify the graph:

• For \( y = -3x - 5 \): Slope \( -3 \), y-intercept \( -5 \) (line goes down from left to right, crosses y-axis at \( (0, -5) \)).
• For \( y = 3x + 1 \): Slope \( 3 \), y-intercept \( 1 \) (line goes up from left to right, crosses y-axis at \( (0, 1) \)).
• The lines intersect at \( (-1, -2) \).

The graph with two lines: one with slope \( -3 \) (decreasing) through \( (0, -5) \), one with slope \( 3 \) (increasing) through \( (0, 1) \), intersecting at \( (-1, -2) \) represents the system.

Answer:

To determine the graph of the system \(

$$\begin{cases} y = -3x - 5 \\ y = 3x + 1 \end{cases}$$

\), we first find the solution (intersection point) by solving the system.

Step 1: Set the equations equal

Since both equal \( y \), set \( -3x - 5 = 3x + 1 \).

Step 2: Solve for \( x \)

Add \( 3x \) to both sides:
\( -5 = 6x + 1 \)
Subtract \( 1 \) from both sides:
\( -6 = 6x \)
Divide by \( 6 \):
\( x = -1 \)

Step 3: Find \( y \)

Substitute \( x = -1 \) into \( y = 3x + 1 \):
\( y = 3(-1) + 1 = -3 + 1 = -2 \)

The solution is \( (-1, -2) \). To identify the graph:

• For \( y = -3x - 5 \): Slope \( -3 \), y-intercept \( -5 \) (line goes down from left to right, crosses y-axis at \( (0, -5) \)).
• For \( y = 3x + 1 \): Slope \( 3 \), y-intercept \( 1 \) (line goes up from left to right, crosses y-axis at \( (0, 1) \)).
• The lines intersect at \( (-1, -2) \).

The graph with two lines: one with slope \( -3 \) (decreasing) through \( (0, -5) \), one with slope \( 3 \) (increasing) through \( (0, 1) \), intersecting at \( (-1, -2) \) represents the system.