Question
Solving the system y = -x + 3 and x + y = 8 by graphing shows that there is no solution to the system. Is
statement true or false? Explain.
(1 point)
• The statement is true, because the lines are parallel.
O The statement is false, because the lines have an intersection point.
• The statement is false, because the lines are parallel.
• The statement is true, because the lines have an intersection point.
statement true or false? Explain.
(1 point)
• The statement is true, because the lines are parallel.
O The statement is false, because the lines have an intersection point.
• The statement is false, because the lines are parallel.
• The statement is true, because the lines have an intersection point.
Answers
GPT-4o mini
To determine whether the statement is true or false, we can analyze the two equations given:
1. \(y = -x + 3\) is already in slope-intercept form (y = mx + b), where the slope (m) is -1 and the y-intercept (b) is 3.
2. To rewrite \(x + y = 8\) in slope-intercept form, we can isolate \(y\):
\[
y = -x + 8
\]
This line has a slope of -1 and a y-intercept of 8.
Since both lines have the same slope of -1, they are parallel. Because parallel lines never intersect, they have no points in common. Therefore, this system of equations does not have a solution.
Now, let's analyze the answer options provided:
1. **The statement is true, because the lines are parallel.** - This is correct because the lines do not intersect.
2. **The statement is false, because the lines have an intersection point.** - This is incorrect; there is no intersection.
3. **The statement is false, because the lines are parallel.** - This is also incorrect; the lines being parallel is the reason the statement is true.
4. **The statement is true, because the lines have an intersection point.** - This is incorrect; as established, the lines do not intersect.
The correct choice is the first one:
**The statement is true, because the lines are parallel.**
1. \(y = -x + 3\) is already in slope-intercept form (y = mx + b), where the slope (m) is -1 and the y-intercept (b) is 3.
2. To rewrite \(x + y = 8\) in slope-intercept form, we can isolate \(y\):
\[
y = -x + 8
\]
This line has a slope of -1 and a y-intercept of 8.
Since both lines have the same slope of -1, they are parallel. Because parallel lines never intersect, they have no points in common. Therefore, this system of equations does not have a solution.
Now, let's analyze the answer options provided:
1. **The statement is true, because the lines are parallel.** - This is correct because the lines do not intersect.
2. **The statement is false, because the lines have an intersection point.** - This is incorrect; there is no intersection.
3. **The statement is false, because the lines are parallel.** - This is also incorrect; the lines being parallel is the reason the statement is true.
4. **The statement is true, because the lines have an intersection point.** - This is incorrect; as established, the lines do not intersect.
The correct choice is the first one:
**The statement is true, because the lines are parallel.**