To address the questions in your uploaded images, let's analyze the equations and the graphical solutions.

Step 1: Analyze the Equations

The system of equations provided is:

1. $y = -2x$
2. $y = -2x + 4$

These are both linear equations in slope-intercept form, $y = mx + b$, where:

• The slope ($m$) for both lines is -2, indicating they are parallel.
• The y-intercepts ($b$) are different: 0 for the first equation and 4 for the second.

Step 2: Graphing the Equations

1. Graph of $y = -2x$: This line passes through the origin (0,0) with a slope of -2, meaning it decreases by 2 units in $y$ for every 1 unit increase in $x$.

   • Key points for this line include:
     • (0, 0)
     • (1, -2)
     • (-1, 2)

2. Graph of $y = -2x + 4$: This line has the same slope (-2) but a y-intercept of 4.

   • Key points for this line include:
     • (0, 4)
     • (1, 2)
     • (-1, 6)

Step 3: Determine the Number of Solutions

Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect. Therefore, there is no solution to this system of equations, as there are no points where both equations are satisfied simultaneously.

Summary of Answers

• The graph shows two parallel lines, indicating they do not intersect.
• Answer to the question: The system of equations has no solution.

Would you like more details on any part of this solution or have further questions?

Here are some related questions for further exploration:

1. What happens if two lines have the same slope and the same y-intercept?
2. How can you determine if two lines are parallel or perpendicular from their equations?
3. What is the significance of the y-intercept in the graph of a linear equation?
4. How would the solution set change if the equations represented intersecting lines?
5. How can you solve systems of equations algebraically instead of graphically?

Tip: When graphing linear equations, identifying the slope and y-intercept can quickly reveal whether lines are parallel, intersecting, or coincident.