Looking at the image you provided, we are tasked with identifying which two equations represent the lines shown in the graph.

Analysis of the Graph:

1. First Line (Positive Slope):

   • It crosses the y-axis at (0, 4), which indicates a y-intercept of 4.
   • It has a negative slope, suggesting the line decreases as x increases.
   • From the graph, the line drops by 1 unit for every increase of 1 unit in the x-direction, giving a slope of -1.
   • Therefore, the equation of this line is $y = -x + 4$.

2. Second Line (Negative Slope, Less Steep):

   • It crosses the y-axis at (0, 2), indicating a y-intercept of 2.
   • This line is less steep than the first one, with a slope of approximately -1/3, as it drops 1 unit for every 3 units in the x-direction.
   • Hence, the equation of this line is $y = -\frac{1}{3}x + 2$.

Conclusion:

The two equations that correspond to the lines in the graph are:

• $y = -x + 4$
• $y = -\frac{1}{3}x + 2$

Therefore, the correct answer is the option that has these two equations, which is the third option.

Would you like any further details on how the slopes or intercepts were determined, or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How can you determine the slope of a line from a graph?
2. What does the y-intercept represent in the context of a linear equation?
3. How does the sign of the slope affect the direction of the line?
4. Can two different lines have the same slope but different y-intercepts? What does that mean geometrically?
5. How can you algebraically verify if a given point lies on a line defined by an equation?

Tip:

When analyzing linear graphs, always check the slope and y-intercept carefully as these are the key components to determine the equation of the line.