Let's analyze the system of equations from the image.

Given:

1. $-x + y = 8$
2. $2x - 2y = -14$

Step 1: Rewrite equations in slope-intercept form

The slope-intercept form is $y = mx + b$, where $m$ is the slope.

For the first equation:

$-x + y = 8 \implies y = x + 8$ So, the slope is $m_1 = 1$.

For the second equation:

Divide the entire equation by 2 to simplify: $2x - 2y = -14 \implies x - y = 7 \implies -y = -x + 7 \implies y = x - 7$ So, the slope here is also $m_2 = 1$.

Step 2: Compare slopes and intercepts

Both equations have the same slope ($m_1 = m_2 = 1$), but the y-intercepts are different:

• First equation: $b_1 = 8$
• Second equation: $b_2 = -7$

Step 3: Conclusion

Since the slopes are the same but the y-intercepts are different, the lines are parallel and do not intersect. Therefore, the system has no solutions.

Final Answer:

$\boxed{\text{No Solutions}}$

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you determine if two lines are parallel or perpendicular?
2. What is the geometric interpretation of infinitely many solutions in a system of equations?
3. How can you use the determinant to solve systems of linear equations?
4. What methods are available to solve systems with exactly one solution (like substitution or elimination)?
5. How does changing the slope or intercept affect the solution of a system of equations?

Tip: When two lines have the same slope but different y-intercepts, they will always be parallel and have no intersection.