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 false, because the lines have an intersection point. The statement is false, 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 true, because the lines are parallel. The statement is true, because the lines are parallel. The statement is true, because the lines have an intersection point.
The statement is false because the lines have an intersection point.
The statement is false, because the lines have an intersection point.
To verify this, let's solve the system of equations step by step.
1. Start with the given equations:
y = -x + 3 ...(1)
x + y = 8 ...(2)
2. To graph the first equation, y = -x + 3, we can plot points on a coordinate plane. We can choose some values for x and find the corresponding y values. For example:
When x = 0, y = -0 + 3 = 3 -> point (0, 3)
When x = 1, y = -1 + 3 = 2 -> point (1, 2)
When x = -1, y = -(-1) + 3 = 4 -> point (-1, 4)
Plotting these points, we can draw a line passing through them.
3. Similarly, let's plot the second equation, x + y = 8:
When x = 0, y = 8 -> point (0, 8)
When x = 1, y = 8 - 1 = 7 -> point (1, 7)
When x = -1, y = 8 + 1 = 9 -> point (-1, 9)
Plotting these points, we can draw another line passing through them.
4. By examining the graph, we can see that the two lines intersect at a single point.
5. Therefore, there is a solution to the system of equations, and the statement "Solving the system y = -x + 3 and x + y = 8 by graphing shows that there is no solution to the system" is false.
The correct answer is: The statement is false, because the lines have an intersection point.
To explain why, let's graph the given system of equations: y = -x + 3 and x + y = 8.
To graph the first equation, y = -x + 3, we can start by assigning different values to the variable x and solving for y. For example, let's choose x = 0:
y = -(0) + 3
y = 3
So, one point on the graph is (0, 3).
Now let's choose another value for x, such as x = 2:
y = -(2) + 3
y = 1
Another point on the graph is (2, 1).
By connecting these points, we can see that the graph of y = -x + 3 is a line that goes from (0, 3) to (2, 1), and continues indefinitely in both directions.
Now let's graph the second equation, x + y = 8. Similarly, we can assign values to x and solve for y. Let's choose x = 0:
0 + y = 8
y = 8
So, one point on the graph is (0, 8).
Now let's choose another value for x, such as x = 2:
2 + y = 8
y = 6
Another point on the graph is (2, 6).
By connecting these points, we can see that the graph of x + y = 8 is a line that goes from (0, 8) to (2, 6), and continues indefinitely in both directions.
Now, let's look at the graphs of both equations on the same coordinate plane. We can see that the two lines intersect at the point (2, 1). Therefore, there is a solution to the system of equations, and the statement "there is no solution to the system" is false.
In summary, the correct statement is: The statement is false because the lines have an intersection point.