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.
A The statement is true, because the lines are parallel.
B The statement is true, because the lines have an intersection point.
C The statement is false, because the lines have an intersection point.
D The statement is false, because the lines are parallel.
The statement is false, because the lines have an intersection point.
To solve the system by graphing, we plot the two equations on a graph.
The equation y = -x + 3 is a linear equation representing a line with a slope of -1 and a y-intercept of 3.
The equation x + y = 8 can be rearranged to y = -x + 8, which also represents a line with a slope of -1 and a y-intercept of 8.
When we graph these lines, we will see that they intersect at the point (2, 6).
Therefore, the correct statement is C. The statement is false, because the lines have an intersection point.
C The statement is false, because the lines have an intersection point.
To solve the system by graphing, we can plot the equations on a coordinate plane and see where they intersect.
The first equation, y = -x + 3, is a linear equation with a slope of -1 and a y-intercept of 3. By plotting this equation on the graph, we can see that it is a line that passes through the point (0, 3) and has a downward slope.
The second equation, x + y = 8, can be rearranged to y = -x + 8. This equation is also a linear equation with a slope of -1 and a y-intercept of 8. By plotting this equation on the graph, we can see that it is a line that passes through the point (0, 8) and has a downward slope.
When we graph both equations, we can see that the lines intersect at the point (2, 6). Therefore, there is a solution to the system of equations. So, option C is the correct answer.
To determine whether the statement is true or false, we need to graph the system and check if the lines have an intersection point. Let's analyze each option:
A) The statement is true, because the lines are parallel.
If the lines are parallel, they will never intersect. Thus, the system would have no solution. To confirm this, we need to graph the lines and check if they are parallel.
B) The statement is true, because the lines have an intersection point.
If the lines have an intersection point, it means they intersect at some coordinated point. Thus, the system would have a solution. To confirm this, we need to graph the lines and check if they intersect.
C) The statement is false, because the lines have an intersection point.
This option contradicts the initial statement. If the lines have an intersection point, it means the system has a solution. To confirm this, we need to graph the lines and check if they intersect.
D) The statement is false, because the lines are parallel.
This option contradicts the initial statement. If the lines are parallel, it means the system has no solution. To confirm this, we need to graph the lines and check if they are parallel.
So, to determine the correct option, we need to graph the lines and see if they intersect or if they are parallel.