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.
(1 point)
Responses
The statement is true, because the lines have an intersection point.
The statement is true, because the lines have an intersection point.
The statement is false, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is false, because the lines have an intersection point.
The statement is false, because the lines have an intersection point.
The statement is true, because the lines are parallel.
The statement is true, because the lines have an intersection point.
The given system of equations is y = -x + 3 and x + y = 8.
By graphing these two equations on the coordinate plane, we can see that the lines do intersect at the point (2, 6). Therefore, there is a solution to the system.
The statement is true, because the lines have an intersection point.
The correct answer is:
The statement is false, because the lines have an intersection point.
To explain why, we can solve the system of equations algebraically and then graph the lines to verify the solution.
1. We have the system of equations:
y = -x + 3 (Equation 1)
x + y = 8 (Equation 2)
2. Solving Equation 1 for y, we get:
y = 3 - x
3. Substituting this into Equation 2, we have:
x + (3 - x) = 8
Simplifying, we get:
3 = 8
4. The equation 3 = 8 is false, which means there is no solution that satisfies both equations simultaneously.
5. When we graph the lines represented by the two equations, we can see that they are not parallel. Instead, they intersect at a single point, which indicates that there is a solution.
Therefore, the statement given is false, and the correct explanation is that the lines have an intersection point.