x + y = 8
y = -x + 3
Solving the system y = -x + 3 and x + y = 8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
True.
When we graph the two equations, we find that they represent two lines that are parallel and will never intersect. Therefore, there is no point that satisfies both equations simultaneously, meaning there is no solution to the system.
x + y = 2 and -x + y = 7? Should the lines for the system be parallel?
No, the lines for the system x + y = 2 and -x + y = 7 should not be parallel.
To determine if two lines are parallel, we compare their slopes. The lines are in slope-intercept form y = mx + b, where m is the slope.
For the line x + y = 2, we can rewrite it in slope-intercept form by subtracting x from both sides: y = -x + 2. In this case, the slope is -1.
For the line -x + y = 7, we can rewrite it in slope-intercept form by adding x to both sides: y = x + 7. In this case, the slope is 1.
Since the slopes are different (-1 and 1), the lines are not parallel.
x + y = 2 and -x + y = 7? Should the lines for the system be parallel?
The graph of -x + y = 7 is incorrect. The lines should intersect.
The graph of -x + y = 7 is incorrect. The lines should be parallel.
The graph of x + y = 2 is incorrect. The lines should intersect.
Both graphs are correct. The lines should be parallel.
The correct answer is: The graph of x + y = 2 is incorrect. The lines should intersect.
The two equations x + y = 2 and -x + y = 7 represent two lines that have different slopes (-1 and 1) and intersect at a single point. Therefore, the lines should intersect, not be parallel.