Does the graph show the system of equations x+y=2 and −x+y=7 ? Should the lines for the system be parallel?
A The graph of −x+y=7 is incorrect. The lines should intersect.
B The graph of −x+y=7 is incorrect. The lines should be parallel.
C Both graphs are correct. The lines should be parallel.
D The graph of x+y=2 is incorrect. The lines should intersect.
A The graph of −x+y=7 is incorrect. The lines should intersect.
To determine if the lines for the system of equations x+y=2 and −x+y=7 should be parallel, we need to examine the equations and the graph.
Looking at the equations:
x+y=2 and −x+y=7
If we rearrange the second equation by multiplying through by -1, we get:
x-y=-7
Comparing the two equations, we can see that the slopes of the lines are different. The first equation has a slope of -1, while the second equation has a slope of positive 1.
Since the slopes are different, the lines will not be parallel. Therefore, the correct answer is:
A) The graph of −x+y=7 is incorrect. The lines should intersect.
To determine if the graph shows the system of equations x+y=2 and −x+y=7, we can compare the equations to the graph. To do this, we need to understand what the equations represent and how to graph them.
The equation x+y=2 represents a straight line in the xy-plane. To graph this equation, we can convert it into slope-intercept form, y=-x+2. This form tells us that the y-intercept is 2 and the slope is -1. Starting from the y-intercept, we can move one unit down and one unit to the right to get another point on the line. Connect these two points to graph the line.
Similarly, the equation −x+y=7 can be rewritten as y=x+7. This form tells us that the y-intercept is 7 and the slope is 1. Using the same process, we can graph this line.
Now, we need to analyze the graph to determine if it represents the given system of equations. If the lines intersect, it means there is a unique solution to the system. If they are parallel, it means there is no solution.
Looking at the graph, we notice that the two lines intersect at a single point. This means that the graph does not represent a system of parallel lines. Therefore, options B and C can be ruled out.
Now we need to determine whether the graph accurately represents the system or not. We can see that the graph shows the point of intersection of the two lines, which matches the solution of the system. This means that the line for the equation x+y=2 correctly intersects the line for the equation −x+y=7. Therefore, the correct answer is D: The graph of x+y=2 is incorrect. The lines should intersect.