To determine which of the given systems of equations can represent the lines shown in the graph, let's examine each option and check if it matches the lines' characteristics based on their slopes and intercepts.

Steps to solve:

1. Estimate the slopes and intercepts of the lines from the graph.
2. Match the slopes and intercepts with the systems of equations provided.

From the graph:

• One line has a positive slope, going upwards from left to right.
• The other line has a negative slope, descending from left to right.

Solution Choices Analysis

Let's analyze each option:

Option A

$5x + 5y = 5 \quad \text{and} \quad -6x + 2y = 2$

• Simplifying the first equation: $x + y = 1$ (dividing by 5), which has a slope of -1.
• Simplifying the second equation: $-3x + y = 1$ (dividing by 2), which has a slope of 3.
• This system has lines with slopes -1 and 3, consistent with one positive and one negative slope.

Option B

$- x - y = 5 \quad \text{and} \quad x - 3y = 6$

• The slopes of these lines are 1 and $\frac{1}{3}$, both of which are positive.
• This doesn’t match the observed slopes from the graph.

Option C

$- x + 3y = 6 \quad \text{and} \quad x + y = 5$

• The slopes of these lines are 1 and -1, which correspond to one positive and one negative slope.

Option D

$- x + 2y = 10 \quad \text{and} \quad x + y = 2$

• The slopes here are $\frac{1}{2}$ and -1, which also correspond to one positive and one negative slope.

Conclusion

From the options provided, Option C is the correct answer as it has one positive and one negative slope that matches the lines' characteristics in the graph.

Would you like a detailed explanation of any steps or have additional questions?

Here are five related questions to explore further:

1. How can we determine the point of intersection for these lines algebraically?
2. How do we find the slope of a line given in standard form?
3. What is the significance of parallel and perpendicular slopes in a system of equations?
4. How can you visually estimate the slope of a line on a graph?
5. What are different methods to solve a system of linear equations?

Tip: For quick slope identification, rearrange the equation into slope-intercept form ( $y = mx + b$ ), where $m$ represents the slope.