Does the graph show the system of equations x+y=2 and −x+y=7 ? Should the lines for the system be parallel? (1 point) Responses Both graphs are correct. The lines should be parallel. Both graphs are correct. The lines should be parallel. The graph of x+y=2 is incorrect. The lines should intersect. The graph of x plus y equals 2 is incorrect. The lines should intersect. The graph of −x+y=7 is incorrect. The lines should be parallel. The graph of negative x plus y equals 7 is incorrect. The lines should be parallel. The graph of −x+y=7 is incorrect. The lines should intersect.

Question

GPT-4o mini · Answer

To analyze the system of equations $ x + y = 2 $ and $ -x + y = 7 $: 1. **Determine the slopes of the lines**: - For the equation $ x + y = 2 $, we can rewrite it in slope-intercept form $ y = -x + 2 $. The slope here is -1. - For the equation $ -x + y = 7 $, we can rewrite it in slope-intercept form $ y = x + 7 $. The slope here is 1. 2. **Compare the slopes**: - The slopes of the two lines are -1 and 1. Since they are different, the lines will intersect. 3. **Find the intersection point** (optional): - Set the two equations equal to each other for the intersection: $$ -x + 2 = x + 7 $$ Combine like terms: $$ 2 - 7 = x + x \implies -5 = 2x \implies x = -\frac{5}{2} $$ Substitute $ x $ back into one of the equations to find $ y $: $$ y = -(-\frac{5}{2}) + 7 = \frac{5}{2} + 7 = \frac{5}{2} + \frac{14}{2} = \frac{19}{2} $$ So the point of intersection is $ \left(-\frac{5}{2}, \frac{19}{2}\right) $. ### Conclusion: - The lines represented by the equations $ x + y = 2 $ and $ -x + y = 7 $ are not parallel; they intersect at the point $ \left(-\frac{5}{2}, \frac{19}{2}\right) $. - Therefore, **the correct response** is: "Both graphs are correct. The lines should intersect."